Think of terrorism in waves. David Rapoport put forth the idea that there have been, so far, four major waves of terrorism starting in the late 19th century. The first was the anarchist wave in the 1880s (mostly in Russia), the second was the anti-colonial wave in the 1920s, new-wave left in the 60s and 70s, and finally today we have the religious wave. Rapoport also states that if this wave follows the pattern of the previous three that it will largely disappear by 2025 that could be followed by a 5th wave that is already starting to happen (and has been for quite a while) (Kaplan, 2008). He also states that there was a founding factor that lead to each wave, though that factor was not always immediate.

This is not to say that terrorism and terrorists have always followed these lines exactly. There have always been groups of various affiliations and goals who committed acts of terror and there are groups from other waves who still exist today (see PKK, Eta, etc).

Some scholars have written that colonialism is the reason for terrorism. A lot of the countries that are generally associated with terrorism and that see a lot of terrorist activities are ones that had been colonised. But not all colonised nations produce terrorism so there must be more reasons. Micheal Mousseau believes that the terrorism Rapoport cites as 'religious' is not actually religious at all but is a form of clientelism (or economic norms theory) where there are groups or individuals fighting to obtain or maintain power and they use religion as a way of recruiting rather than the main reason behind the goals. Again, a lot of the countries that see terrorist activities are ones that are fairly new in terms of historically existing. There is still a rush for power and there is generally a positive correlation between terrorism and unstable governments (Campos and Gassebner, 2009).

If you asked five different experts you would get five different answers. A lot of this disagreement comes from the fact that terrorism really has no set definition and is more of a 'we'll know it when we see it' type of thing. There are certain actions that make something terrorism but the underlying reasons are not always the same. There are so many variables present in why terrorism follows these waves that it is difficult to fit it all into a nice little box. If that was possible it would be much easier to combat.

Campos and Gassebner, International Terrorism, Political Instability and the Escalation Effect, No ,4061

Kaplan, J Terrorism’s Fifth Wave: A Theory, a Conundrum and a Dilemma, Terrorism Research Initiative, Vol. 2, No, 2, (2008)

Mousseau, M, Market Civilization and its Clash with Terror," International Security, Vol. 27, No. 3 (Winter 2002-2003)

Rapoport, D, The Four Modern Waves of Terrorism

Let me try to answer this in specific terms rather than in reference to cyclical movements or what have you. Apologies in advance if its a bit long winded, I find the story to be interesting.

Islamism as an ideological political movement is a relatively new phenomenon. While there are any number of intellectual precursors to contemporary political Islam (Ahmed Ibn Hanbal, Al Ghazali, Ibn Taymiyyah, Ibn Abd Al Wahab) the connection of Islam, political ideology, the foundation of an actual political movement, and an internationalist pan-muslim outlook really first emerges under Hassan Al Banna, who founded the Muslim Brotherhood in Egypt in 1928.

The Muslim Brotherhood, as a key part of its program, and inspired by fascist and communist movements in Europe, organized its young men into paramilitary cadres, the "Secret Apparatus." In the 1930s these paramilitaries engaged in street battles with the youth paramilitaries of other Egyptian political movements, the communists, the Wafd, etc.

The 1936 Palestinian revolt, and eventually, the 1947 civil war in Palestine, provided an opportunity and the motivation for an armed campaign. The Brotherhood sent military forces into Palestine and stockpiled weapons within Egypt. Eventually the brotherhood turned its sights on Jews within Egypt and launched a bombing campaign. The Egyptian government believed things were getting out of control and tried to legally restrict the Brotherhood, which in turn mounted a campaign against the government, killing the prime minister, which resulted in the death of Hassan Al Banna in a reprisal in 1948.

When the Free Officers came to power under Gamal Abdel Nasser there was an initial moment when it looked as though the Brotherhood would have a special role in the new Egypt. That idea fell apart, and the Brothers made an attempt on his life.

Here we get to the interesting part, in terms of terrorism. As far as I'm aware, the plot to kill Gamal Abdel Nasser marks the first proposal by an Islamic terrorist group to launch a suicide bombing.

The idea was rejected by the brothers as being too extreme. Instead they tried to shoot him.

The response from the government was massive and overwhelming reprisal. The brothers were imprisoned, tortured, and quite often executed.

It was in one of Nasser's prison that Sayyid Qutb wrote Milestones, arguably the foundational text of international terrorism and Jihad. Milestones put forward the idea that western nations were "jahili," living in a state of decadent pagan ignorance, and using the same term for the paganism that the prophet Muhammad cleansed from Arabia by violence. What's more, by cooperating with the west and preventing the enactment of the Sharia, the regimes of the Islamic world were actually, themselves, in a state of jahilia as well.

Publishing the book from prison in 1964, Qutb would be put to death in 1966.

In a sense you could conclude your question at this point, because the short answer is TL;DR that it took a few decades from 1966 for Qutb's doctrine to spread. But there's a good deal more to it than that, which I'll touch on here.

Going backwards in time a bit to a region of the world that has become perhaps most associated with terrorism, let's talk about AfPak.

Around the same time that Banna founded the Muslim Brotherhood in Egypt there were similar revivalist movements in Southwest Asia. The Khilafat movement attracted much attention from the Muslims of India in a brief but inspiring period in the early 1920s. Islamic nationalism as a whole had a much broader appeal, starting with Muhammad Iqbal, the Muslim League and most famously espoused by Muhammad Ali Jinnah.

But the Khilafat movement captured the imagination of Abul Ala Maududi. Maududi founded the Jaamat-e-Islami, who also laid out its ideology.

It was Maududi who created the modern conception of the "Islamic State" based on Shariah. Without getting into the 20 year rule, while people point to Saudi as the model for the Islamic State, that's true only in its strict application of the Sharia. In its style and argumentative method, it's appeal, radical islam today owes far more of its ideology to Maududi's thinking.

To skip ahead a bit, the JI was essentially given the kind of outsized role Muhammad Zia Ul Haq (r. 1977-87) that the MB had hoped for under Nasser.

It was under Zia that Pakistan launched a program of building and funding the infamous Madrassas that churned out Mujahideen fighters from Afghanistan to fight the soviets, again with the JIs vision of Islam as a key part of the curriculum.

That's skipping ahead a bit though.

The foundation of Israel likewise played an enormous role in the eventual creation of modern Islamist terror. The defeat of the Arabs in 1948 led to Saudi Arabia, in particular, playing an outsized role in the funding of all kinds programs for Palestinians and other Arab groups. I think this is perhaps better known, but Saudi Arabia became a major source of money for funding hard-line Islamic institutions abroad as well as a destination for that kind of learning within the kingdom itself.

As I think I've probably exhausted myself at this point I think I would just point out how key the year 1979 was in this. That year saw 1. Saddam Hussein's rise to power in Iraq. 2. Ayatollah Khomeini's overthrow of the Shah in the Islamic Revolution. 3. The Soviet Invasion of Afghanistan. 4. The signing of peace between Israel and Egypt.

Those four events set in motion much of what set the stage for the emergence of today's various terrorist movements, including Al Qaeda, Hezbollah, the Taliban and the Islamic state, among others.

Sources:

For the early history of the Muslim Brotherhood, The Society of Muslim Brothers by Richard Mitchell. For Islamist ideology, Princeton Readings in Islamist Thought, Texts and Contexts from al-Banna to Bin Laden. A history of world events leading up to the foundation of Al Qaeda and international terrorism see Ghost Wars by Steve Coll. For a first hand account of the early days of what would become the War on Terror, including America's first major experience of Islamic terrosim, the bombings in Beirut, see See No Evil by Robert Baer. For a general overview of the Middle East, including the rise of Islamism and Islamist terrorism, see Eugene Rogan's The Arabs: A History.

Islamic Terrorism increased greatly throughout the 1980's because of a radical change in the relationship between in Islamic militancy and the Islamic/Arab State system. Long story abbreviated, Islamic militancy was largely coordinated with Arab/Islamic states from their independence until their relationship began to change in the mid 1970's when the PLO began to lose favor with their Arab host governments, when the numerous Arab/Islamic states began to alter their relationship with the West and Israel, and when national Arab government's began stopping direct cooperation with Islamic militancy group. When the latter happened, the militant groups flourished into huge transnational networks, not directly supported by national governments, but supported by wealthy donors within those former countries, who have grown progressively wealthier since the 1980's.

After the creation of Israel, and the independence of the Arab states in the 1940's, the Arab states had coordinated with Palestinian fedayeen to attack Israel. Through 1970-1979, militant action by the PLO destabiilized Jordan, Lebanon, and the subsequent Muslim Uprising in Syria, the Arab nationalist movements outside the established national governments fell out of favor with the Arab autocrats, and the militant groups were banned and marginalized.

In 1979, numerous political developments gave an opportunity for Arab governments to "free" themselves of active Islamist groups by "exporting" them to fight outside their home countries.

In 1979, the Grand Mosque Siege profoundly changed the political and religious culture of Saudi Arabia. Driven by radical Islamists who saw the King as Westernizing the Muslim Kingdom, the siege was the greatest uprising against the Westernizing of the middle east, in general. In the aftermath of the siege, the King gave increased powers to the religious establishment in hopes of combating extremism by giving them a central role in Saudi society. The aftermath of the siege pushed Islamic movements in the heart of the Islamic world into direct conflict with westernizing Arab governments, rather than as allies.

Also in 1979, Egypt and Israel sign a peace treaty, the first treaty between an Arab state, and the first Arab country to recognize Israel. This marked the ending of state sponsored conflict between Israel and its Arab neighbors, and close cooperation between the Palestinian nationalist movement and the Arab governments. This change promoted the transnational Palestinian and Islamist movements to the first line in the fight against Israel. From here on out, the Arab states would become more hostile to elements in their country who wished to initiate conflict with Israel, while those movements began to operate further outside the framework of their host states.

Lastly, at the close of 1979, the Soviets invaded Afghanistan, initiating the recruitment drive of Islamist fighters, and creation of a transnational movement of Islamist supporters who came from all over the Islamic world to fight the Soviets, learn from the mullahs and Islamic schools that sprouted up in the tribal lands of Pakistan and Afghanistan. The movement of fighters was supported by many Islamic governments who used the conflict as a kind of pressure valve. It was considered an opportunity to rid their countries of violent and revolutionary Islamist elements in their countries and export them to Afghanistan, thereby sparing them domestic security threats. The calculation that their home countries would be safer by exporting their militants to fight in Afghanistan had the harshest effect on the creation of huge terrorist networks that exist today.