It's hard to say how common it was, although blood feuds between tribes certainly occurred and individuals did kill. One of Muhammad's changes was to declare that blood money would be paid after a murder; Muslims were not supposed to kill for revenge. However, if they insisted on blood, they could only take vengeance on the individual who committed the crime. Before this, it was legal to kill any male member of the offender's tribe (blood money also existed in pre-Islamic Arabia; Muhammad emphasized it, however). Personal insults were certainly a common cause of strife, and personal honor was personally and socially important to the people of the day. The nature of insults could run from the personal ("You worthless dog!") to the familial ("A thousand curses on the womb that bore you!") to the tribal ("May fire consume every person who shares your blood!"). Just as in the modern day, such insults could--and did--often lead to violence.

There is a story told that once, when Muhammad was praying at the Ka'aba (very early on in his prophetic career) and some of his detractors (specifically a man later known as Abu Jahl أبو جهل) came and threw a bucket of blood and entrails on him. Muhammad's uncle Abu Talib, and his friend Hamza, took vengeance, but did not kill. Accounts vary from them beating the offenders bloody, or rubbing the entrails in their faces or beards. This could be an apocryphal story, however, as details of Muhammad's life were generally not recorded until a century after his death. Apocryphal or not, the story itself is repeated because it shows Muhammad responding in a way meant to illustrate the right way to do something, replacing the former.

Violence was not the only way disputes were settled. Poetry was considered a very manly pursuit, and sometimes individuals would fight back and forth via poems--sort of an ancient version of today's freestyle rap battles.

Violence was also accepted through raids on other tribes and caravans; this became a method of redistributing resources in a land where they were scarce.

One the legacies of Muhammad's life was his creation of a 'super-tribe'; the ummah, or Muslim community. One's allegiance to God and religion was meant to take the place of tribal loyalties; by the end of his life he had unified the Arabian peninsula, arguably for the first time ever. Sadly, tribalism is still a powerful force in Saudi Arabia today.

Of course today we often read of violence and murder in the Islamic world, directed at those who insult Islam or Muhammad. As the Qur'an and ahadeeth often support this sort of thing, it would be wrong to say that Muhammad's teachings were pacifistic. It should be noted that his judgements against foes were harsher after the migration to Medina, when the ummah grew large enough to have some measure of power.

Sources:

Bloom, Jonathan and Shiela Blair (2002) Islam: A Thousand Years of Faith and Power

Hourani, Albert (1992) A History of the Arab Peoples

Lapidus, Ira (2002) A History of Islamic Societies

Lings, Martin (2006) Muhammad: His Life According to the Earliest Sources

NB: On mobile at the moment; if you require more details or have follow up questions I'd be happy to address them when I get to my desktop.