There were. A lot.

Western Europe had two main issues with terrorism: ethnic nationalists, and far-left terrorism. For example, Britain had a serious problem with Irish nationalist terrorist groups, primarily the IRA, throughout the entire 20th century, especially the 1970s and 1980s. Spain had a similar problem with ETA, the Basque terrorist group. Meanwhile, Germany and Italy had problems with far-left terrorist groups like the Red Brigades. Turkey, on the other hand, had problems with far-right terrorists (and also Kurdish seperatists).

Now, in the Middle-East, Africa and Latin America, there were a lot of what would be labelled terrorist groups these days. The difference then was that they would often seize control of the state or large portions of the state, and thus were labelled as rebels rather than terrorist groups. The reason it might appear there were less terrorists is not that states were more stable, but that they were less stable. Governments being seized by far-right or far-left insurgent groups, or different insurgent ethnic groups, was a lot more common than it was now.

Finally, unlike today, the decolonisation process of the various Empires was rife with terrorist activity, but again, what we'd label "terrorism" today was generally labelled "rebellion". For example, the bombing the King David hotel by Zionists trying to drive off the British rulers of Palestine.

The main difference today are the rise of Islamic fundamentalist terrorism as a guiding ideology (though this was a problem in the 20th century, just less of one: it's not something that came out of nowhere on 9/11), with the fall of nationalist terrorism (though it should be noted that nationalist terrorism is still a thing), and the fall of far-left terrorism (which is also still a thing: Maoist rebels in south-Asia is still a problem). Far-right terrorism is still around though (for example Anders Breivek). Islamic fundamentalist terrorism also uses suicide bombers much more than other kinds of terrorism, and this is a very effective tactic for causing mass civilian casualties. Finally, the western media reports terrorism around the world a lot more than it did before 9/11, which makes it seem like terrorism is more common.

It's very important to remember that while "terrorism" does have a specific definition, what people consider terrorist groups is often related to the politics of the groups involved. I, a British man, define the IRA as a terrorist group; my Irish republican friend considers them a rebel group.

I think you have a great misunderstanding of Terrorism in the 20th Century as compared to the 21st. Of course with current events such as 9/11, the British bombings in 2005, Paris in December 2015, it's fresh and current in our minds, but quite honestly, terrorism was just as, if not more widespread in the 20th Century.

As one example in America, in September of 1920, 38 people were killed in a bombing on Wall Street in New York City. In those days, Wall Street was still the physical location of the major trading companies as well as banks, so it was a physical attack on the American financial sector. This was the deadliest attack since the Los Angeles Times bombing in 1910 which killed 21. Both of these were Labor/Anarchist tied bombings. These attacks were the most notable events that went with mail bombings, assassination attempts on political figures as high ranking as Congressmen and the U.S. Attorney General, bank robberies (of Sacco and Vanzetti fame), anti-war bombings, etc. under a movement generally called "Galleanist Anarchists." These activities which were so numerous and widespread that it led to the Immigration Act of 1918 which curtailed the admittance of people of certain (Anarchist/Communist) political sympathies. These acts led to the rise of J. Edgar Hoover who helped coalesce the FBI into its present form. Hoover would go on to give support to such reactionary measures against the Anarcho-Socialists of the time that they resulted in numerous deportations of people such as Emma Goldman.
Mind you, these events are just up to about 1925 in the United States alone. In 1950 President Truman was attacked by Puerto Rican nationalists but survived.

Now we come to the 1960's and the world literally exploded. You have the growth of the Symbonise Liberation Army, The Red Brigades, the Palestinian Liberation Organization, New People's Army, Umkhonto we Sizwe, FARC, The Weathermen, Bader-Meinhoff Complex, Hamas, the KKK, Shining Path, Black Liberation Army, Japanese Red Army, Basque Separatists, the Irish Republican Army and Sein Fein, Lehi and the Black Hand, German Red Army Faction, Tamil Tigers....

You get the idea.

The period from the 1960's through about the late 1980's were extremely driven by terrorism focused on either Communism, Nationalism, or Sectarian divides. What partially was so shocking about 9/11 is that it was the first successful domestic American hijacking since 1983!

In fact, hijackings and attacks through the 1960's and 1970's gave rise to many of the worlds first notable "anti-terror" units such as Germany's GSG-9 (a reaction to Black September and the 72 Olympic attacks), Belgian DSU, Indonesia's VAT 69, American SWAT teams in major cities, etc.

This is all going just off the top of my head and is very American focused, but terrorism in the 20th Century was a driving force in geo-politics and domestic policy for much of the world, shaping its police forces, its foreign policy, and its cultural values. On a relativistic and probably Amero-centric view (as well as potentially breaking the 20 year rule), terrorism today is not nearly the issue it was fifty or sixty years ago.

I'm not sure where you get the notion that there weren't terrorist organisations in the 20th century from - just off the top of my head there was the IRA, ETA, the Tamil Tigers, Sendero Luminoso, the Mujahideen, etc. (Whether you consider these groups "terrorist" or not is a separate matter, of course). And these are just some of the large scale groups: there are also the Red Army Faction, the Weathermen, GRAPO, and a whole host of other, smaller groups operating.

Forms of terrorism have been around since warfare has begun, but the frequency and publicity has been slightly larger since the 1960s. Like other comments there are numerous groups who have been operating before this time frame.

One thing to get straight is that the point and focus of most terror groups throughout history is not to inflict mass causalities on civilian populations because terrorist groups fight on their own land, in their own backyards so inflicting mass casualties against their own populations would be counter productive. The point of a lot of terrorism to bring in or promote a political position. For example the IRA in Ireland didn't mostly target civilians in Ireland rather they targeted British Army positions and bases that sometimes had civilians in them. Here is a website that details some of the attacks and deaths in Ireland during this time.

You also mentioned during WW2, there were many 'terror' groups in Europe during this time, but are now considered freedom fighters because one man's terrorists are another's freedom fighter. These groups include the French Resistance and Polish resistance fighters.

These are not considered terrorist groups now but the definition of 'Terrorism' is the use of violence or other means to influence political matters. Even though these groups are not terrorists now, they fit the definition and you can learn a lot about current groups from them.

The French resistance is a very interesting group. They're main purpose was to combat the Vichy government established after the German invasion. The groups did a lot in France during the occupations such as destroying rail lines, helping shot down ally pilots return to safe territory, they published underground newspapers, intercepting and sending intelligence back to the allies and helping the D-Day invasions by working on the ground weeks before the operation. The French resistance got a lot of help from the British and Americans because these two countries sent men into occupied France to train the resistance soldiers to combat the Germans. The American OSS, which was a major factor in helping the Resistance and winning France is the pre-courser to the CIA. Though if it wasn't for the British MI6 for training to OSS officers and helping the resistance they would have had no chance in defeating the Germans.

The largest Polish resistance force was called Armia Krajowa, which translates to 'Home Army', they were loyal to the Polish Government in exile. Some estimates put their numbers to well over 200,000 throughout Poland. Though not all were fighters logistics is what wins wars. The many resistance groups in Poland were a lot more ruthless than their French counterparts because there were dramatically more German troops on the Eastern Front than on the western. The most famous battle these groups fought was the 'Warsaw Uprising'. This was a two month long battle which the Resistance ultimately lost but was a major hit onto the Germans in the city. Around 85% of the city was destroyed, over 100,000 civilians were killed and forced the Germans to withdraw from the city. Which led to the Soviet occupation of the city. Sadly after the Soviets invaded and annexed Poland from the Germans the Home Army was disbanded because the Soviets saw them as a direct threat to their rule.

I'll write some more examples, clean it up and add sources after class.

Some great answers here, but I'd just like to add that Terrorism in the 21st century is perhaps perceived as being a bigger problem because of the much wider media coverage it now receives. Through the use of television coverage then this was, for example, played upon by the Palestinian terror group Black September during the Munich Olympic Massacre in 1972. With the advent of the internet then the audience only gets bigger, and now we have ISIS, who are the first real example of a terror organisation who actively promote their terror acts through the use of the internet.