Someone who studied this in more detail or to a higher level than me could give you more detail, but seeing as no-one else has answered this yet I thought I'd give tackling this ago. I studied this for my BA in History a year ago and it was one of my favourite topics.

The foundations of Islam and the Islamic world are in a large part built on the ruins of the Romans, but the issue is complicated and the links not always straightforward.

The "Roman" influence on Islam/Islamic culture comes indirectly, through Constantinople, the capital of what is often known as Byzantium, sometimes referred to as the "Second Roman Empire". It's actually a lot more complicated than that; Constantinople was originally part of the greater Roman Empire, before it split in 285 CE into a Western (Capital: Rome) and Eastern (Capital: Constantinople) half. While the "fall of the Roman Empire" is often dated to 476 CE, Constantinople - the Eastern half - remained.

However, the extent to which Byzantium was "Roman" or not is subject to a lot of debate. For one thing, it was predominantly Greek speaking. It was an often turbulent, Christian society, with pagan vestiges increasingly coming under attack, and its Chistrianity looked very different from that practiced in the West.

During this period, Constantinople was engaged in a colossal struggle with another empire, the Persians. This involved long periods of intermittent warfare. The powers vied for dominance over the known eastern world, including influence over smaller client kingdoms.

Enter the Arabs. These were a mostly nomadic people living in the Arabian desert. However, during the sixth/seventh centuries (about 500CE/570CE historians recognize that they were becoming wealthier, and cities such as Medina and Mecca were becoming important trading hubs. As the importance of the Arabs grew, their land and culture was subject to the interest of the Byzantium and Persian superpowers. Arabs came into contact with and were aware of Byzantine/Persian culture - including Christianity.

Almost immediately following the rise of the Prophet, both the Persian Empire and Byzantium went through an extremely difficult period where both Empires were devastated by warfare, economic near-collapse, and (especially) plague. I am massively over-simplyfying here but this is the jist of it.

So we are now left with two weak, vulnerable superpowers in the East. Enter the Arabs: powerful, not heavily hit by the plague, newly (partially) urbanised, still generally adhereing to syncretistic, somewhat disparate animistic religions. This is where I can finally get about to answering your question.

Firstly, the religious philosophy of Islam (note that I am speaking as a non-Muslim here and believers may have very different takes on this). It has been suggested that Muhammad and his followers by influenced by the power of monotheist religions they had witnessed. The two most powerful cultures that the Arab world had been in contact with worshipped one God. This new religion was a way in which the Arabs, too, could adhere to a belief system that was universal and all-encompassing.

There are other aspects of Islam which have interesting similarities to Byzantine Christianity, too. Take the concept of holy war: the Eastern Roman Empire had for a long time viewed their struggle with the Persians as a holy war. Their Christianity was heavily inflected with ideas of warfare, even apocalyptic warfare. Take also the extremely evangelical outlook of Early Islam: the world was literally divided into two "houses", of the believers and the unbelievers, and the early Islamic conquests seem to have been predicated on the understanding that the whole world would be converted to Islam. This too is very similar to the evangelical outlook of the Eastern Roman Empire and the way they viewed believers vs. non-believers.

Secondly, the Islamic conquests - by which armies led by Muhammad, and later his followers, moved out of the Arabian desert and with almost astonishing rapidity took over North Africa and the Middle East, and Persia, was a direct conquest of the Eastern Roman Empire. Persia was entirely taken over, and only a rump of the old Eastern Roman Empire remained. Once conquered, these "Roman" areas (such as Syria, Libya, Palestine) were adminstered relatively liberally, in that the original inhabitants were allowed to continue to practice Christianity as long as they paid a tax to Muslim conquerors. The process of Islamification was generally quite slow, and sometimes involved a far higher degree of local accomodation and co-operation than often appreciated. For example, in Egypt, the Church was put in charge of collecting the taxes of Christians by the Islamic conquerors.

There's a lot more I could go into but I should probably leave this to someone who's got a PhD in this or something as it's complicated (and fascinating). As you can probably tell I've been massively simplifying things. There are all sorts of interestering links, such as the fact that the followers of Muhammad may well have included Christians and Jews, for a time, before the idea of Islam as a religion that was mutually incompatible with other montheistic faiths cystallised. For now, I suggest reading Dr Peter Sarris's Empires of Faith - The Fall of Rome to the Rise of Islam, c.500-700, which is excellent, very readable, and covers exactly this topic.