Politically the Fertile Crescent of the early 7th century was split right in the middle between two large empires, Rome and Persia, and it had been so for more than six centuries. Nevertheless there were some significant cultural similarities throughout the hole region cutting right across this political divide. For one thing the dialects spoken by the majority population from Palestine all the way to the Persian gulf were variants of the Aramaic language. This contrasted with the languages spoken by the elites of both empires, Greek and Latin for the Romans and Middle Persian for the Persians, which were probably heard mostly in the cities of the region and a lot less often in the country side. Otherwise there were Arabic speakers mostly in the form of nomadic and semi nomadic tribes in the steppe regions but also among the settled population. They formed a kind of bridge between both empires and the tribal societies to the south.
Also people on both sides of the frontier were largely christian. But while Christianity was the dominant religion inside the Roman Empire it was only tolerated by the Persians who’s political elite mostly practiced Zoroastrianism. Nevertheless the majority of Persia’s Christians lived in Mesopotamia. Beside the Christians there were large and ancient jewish communities in Mesopotamia as well as in larger Syria. Although they were a politically disadvantaged minority in both empires they still had a significant cultural impact. In fact the jewish communities of the Fertile Crescent were probably the largest and most influential of their time. It was here where the foundations for the classical jewish tradition were laid with the compilation of the Babylonian and the Palestinian Talmud throughout Late Antiquity. There were also other smaller religious groups like the Manichees, Mazdakids or polytheistic cults spread throughout the hole region. Especially the Arabs were split into Christians and Polytheists.
Now the political divide between the Roman and the Persian Empire was not totally without impact on this cultural continuum. For example at least since the 6th century the Arab tribal groups of the region were largely organized into two large confederacies that fought on behalf of one of the empires, the Lakhmids on the side of the Persians and the Ghassanids on that of the Romans. The christian religion of the aramaic speaking population was also split along political lines. In the 5th century the church of Persia had declared itself in favor of a theology that was considered heretical by the official Roman church and by that announced its independence from the west. This christian denomination still exists in the form of the Assyrian Church of the East also known under the designation given to them by their theological adversaries in the West, Nestorians. With this move the Persian church hierarchy made sure that it wasn’t perceived as some kind of fifth column for the Romans. In this regard they were quite successful and their head, the Katholikos of Seleukeia-Ctesiphon, was an important dignitary at the Persian court. The same was true for the leader of the jewish community, the Exilarch.
By the 6th century this clean divide between the Christians of the Fertile Crescent got a little muddied again. Further theological disputes had lead to another split inside the Roman church. Large parts of Syria now favored the Miaphysite form of Christianity which was mostly suppressed by the emperors of Constantinople. While the emperor’s side could hold its ground in the large hellenised cities the Miaphysites dominated much of the country side. The later also had no difficulties traversing the frontier into Persia and began converting people there as well. By the beginning of the 7th century there was a significant community of Miaphysite Christians inside Mesopotamia. Nestorians and Miaphysites also competed for the allegiance of the Arab confederations, with the former dominating within the Lakhmids and the later within the Ghassanids.
While both the Romans and the Persians had originally come as conquerors to the Fertile Crescent by Late Antiquity the region was far from peripheral to both of those empires. Important political and cultural centers were to be found here. On the Roman side the pride of place probably belongs to Antioch in Northern Syria, modern Antakia. After the capital Constantinople and Alexandria in Egypt it was the largest city in the Roman East by far. Its bishop held the honorary rank of patriarch and was one of the most senior clerics in the whole Levant. In the 4th century it had also functioned as an imperial residence but by the 7th century it had been centuries since an emperor had visited the city. Nevertheless it was still the seat of several important dignitaries like the magister militum and the praefectus praetorio of the orient and the elites of the city owned large estates throughout the whole of Syria and were politically well connected. Much smaller in size and political importance but surpassing Antioch in religious prestige was Jerusalem in Palestine. Its bishop also held the rank of patriarch and the city was, for obvious reasons, one of the premier pilgrimage destinations in all of Christianity. On the Persian side the most important urban center by far was Seleukeia-Ctesiphon at the river Tigris, a huge agglomeration of several cities dating back centuries. It served as the capital city of Persia and the primary residence of the king of kings. Both Antioch and Seleukeia-Ctesiphon were foundations from hellenistic times when the Seleucid Empire had ruled over the entire Fertile Crescent. The connection between the two cities got strengthened again in the 6th century when the Persian king Khosrow I Anushirwan sacked Antioch and deported large parts of its population to found a new suburb for his capital which he called Veh Antiok Khusrau (“the better Antioch of Khosrow”).
Other cultural centers attracted people from both sides of the frontier. An important example was the pilgrimage site of Resafa near the Euphrates. Its patrons, the military saints Sergios and Bacchos, were hugely popular with the Arab tribes on the Roman and Persian side as well as in the south. Both emperors like Justinian I and kings of kings like Khosrow II Aparvez made costly donations to the sanctuary.
I hope all of this shows that the Fertile Crescent of the 7th century was at the same time culturally hugely diverse and deeply influenced by its political division as well as strongly interconnected at the same time. This was the cultural heritage that the Arabs attained when they conquered the region. Plenty of things would change afterward. For example Antioch and Seleukeia-Ctesiphon lost much of their importance while new cities like Basra and Kufa in Mesopotamia rose up. Islam replaced Christianity and Zoroastrianism as the religion of the elites. After a time Arabic would replace Greek and Middle Persian as the language of state. At the same time there were plenty of continuities. The Umayyad rulers would build impressive palaces and religious shrines in old centers like Jerusalem or Resafa. The Nestorian, Miaphysite and Jewish dignitaries held important positions inside the court hierarchy of the Abbasids like they had done under the Persians. Late Roman and Persian elite culture had a huge influence on early Islamic civilization.
Edit: I probably should add some sources:
William H. C. Frend, The Rise of the Monophysite Movement (2008)
Elizabeth Key Fowden, The Barbarian Plain. Saint Sergius between Rome and Iran (1999)
Richard E. Payne, A State of Mixture. Christians, Zoroastrians and Iranian Political Culture in Late Antiquity (2016)
Peter Sarris, Empires of Faith. The Fall of Rome to the Rise of Islam 500-700 (2011)
Follow-up: What do we know about the pre-Islamic "pagan" religion of what is now Arabia?