I was thinking about this and realized I couldn't really place any major civilizations to this area at the time immediately preceding the Muslim conquests... It's far too late for the Assyrians or Babylonians, but well before any Turkic people come down from Central Asia, so no Ottomans or Safavids. The only major one I can think of is maybe the Persians, but I guess I don't know enough about their time periods to be sure.
Basically I'm wondering what the general political structures were in that time period throughout the Middle East, not just Persia or the Levant, but Arabia and Anatolia and North Africa. What would the average person's social/religious/political fealty be?
What was the political structure of the Middle East immediately before the Muslim conquests?
Monarchy.
... What, you want more than that? Okay then... but judging by your (correct) assessment that neither the Assyrians (since the 6th century BC) nor Safavids (before the 16th century AD) were a political force of any magnitude whatsoever, I think I have to sketch out a timeline. But first, to answer your question directly: The region was dominated by the (Eastern) Roman Empire and the Sasanian Empire. This map illustrates the situation in 600 AD, shortly before the conquests. Note that unlike what the map indicates, the Ghassanid kingdom should be properly regarded as a Roman vassal in this period. As you can see, the tribes of the Hejaz and the inner Arabian peninsula are among the few polities not dominated by either Persian or Roman politics. Meanwhile, the vassal kingdoms of the Peninsula served as a source of mercenaries for the Roman-Persian wars, which has recently come to be regarded as important context for the eventual Arab conquests (for example, Arabian armies from the east of the peninsula raided the weakened Sasanians even before they were part of the early caliphate).
Indeed this seems to be a recurring theme in the history of empires - at their borders, client and allied states grow more and more powerful and influenced by imperial ideology until they eventually grow powerful enough to invade the empire at a weakened time. This can be seen not only in the Arab Conquests, but also in Alexander's conquests, various Sino-Turkic dynasties, the Gothic kingdoms, the Seljuk conquests, and if we were to stretch this idea well beyond its breaking point we could perhaps even shoehorn Carthage and Rome themselves into a similar framework at the edges of the Graeco-Persian world.
Anyway, religions. The most common religion west of, and maybe even in, Mesopotamia at this time was Christianity (with many variants and "heresies" still around). In the Sasanian Empire, the most common religion would have been Zoroastrianism, both of the "official" kind and in more diverse forms, especially in the east, where Manichaeism (which I usually describe as a syncretism of Gnostic Christianity and Buddhism situated within a quasi-Zoroastrian framework) was also prominent, and judging by certain Persian sources there must also have been non-trivial numbers of Hindus and Buddhists there.
So, just let me draw up a quick timeline of the major polities to give you an idea of the 1000 years before the Arab conquests, since I know how hard it is to keep track of this stuff:
In the mid-5th century Cyrus II "the Great" of Anshan conquers the last Babylonian kingdom. His sons Cambyses and Bardiya are (possibly in the case of Cambyses, definitely in the case of Bardiya) assassinated by the nobleman Darius of the Achaemenid clan, who usurps the throne, crushes the resulting rebellions, marries Cyrus' daughters Atosa and Artystone and claims a spurious distant cousinship with Cyrus, beginning the true line of the Achaemenid dynasty, the realms of which encompassed the entirety of what we today call the Middle East, except for the southern half of the Arabian peninsula.
In the 330's BC, Alexander III of Macedon invades the largely intact Achaemenid realms and conquers them. On his death, a succession crisis occurs, and the realms are eventually [split.](https://en.wikipedia.org/wiki/Macedonia_(ancient_kingdom)#/media/File:Diadochi_LA.svg)
A chieftain named Arsaces establishes a kingdom in the northeast of the Seleucid Empire illustrated above in the mid-3rd century BC. This dynasty gradually expands, and Mithridates I conquers most of the Seleucid Empire east of the Euphrates between 171 and 139 BC. See this animation
The parts of the old Macedonian Empire west of Mesopotamia are conquered by Rome in the 1st century BC, and vassalizes Armenia, ruled by an Arsacid king.
In the first two centuries AD, territory in Mesopotamia, Syria, and Anatolia switches hands between Rome and Parthia/The Arsacids numerous times, while Parthia loses much territory in the east to the Kushan Empire (these are also illustrated in the above animation).
In 224, the Persian vassal king Ardashir rebels against the Parthian Arsacids, conquers them, and re-conquers the lost Eastern territories, establishing the Sasanian Empire.
Over the following 400 years, Rome and the Sasanian Empire struggle intermittently over such matters as hegemony over Armenia (still ruled by Arsacid kings). For much of the time the Romans essentially pay off the Sasanians to not have to go to war, and the Sasanians for this reason nominally regard the Roman Empire as a tributary client state.
The war to end all wars erupts in 602 and lasts until 628. Though the Sasanians nearly push the Romans out of Asia entirely their gains are reversed by Heraclitus, and the war ends in a territorial status quo when the Sasanian king Khosrow II is overthrown by his son Kavadh II, who sues for peace. The Sasanian Empire erupts into a four-year civil war, eventually resulting in the appointment of figurehead king Yazdegerd III, who would reign for about twenty years and lose the war-torn and destabilized empire to the Arab invasion.
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