I wrote a little about the history of the modern Syriacs/Assyrians. The relevant paragraph is probably:

Are they really Assyrian? Well, there's really nothing else to call them. They live in a region that historically has been a mix of Kurdish, Arabic, and Turkic speakers, along with minority languages Armenian, Greek, and Syriac. Unlike the Christians in their traditional homeland, they neither descend or speak the languages of the Greeks or the Armenians. They are organizationally separate from the Arab Christian churches, which are Greek/Eastern Orthodox or Catholic, with Protestant now as well, nor do the communities in Turkey and Iran speak Arabic. They are not descendant from the Turks, nor do the communities in Syria and Iraq speak Turkish. This is why Maronites of Lebanon often call themselves Phoenician: they may speak Arabic now, but they are not descended from the Arab invaders. Coptic Christians, similarly, sometimes claim to be ethnically the same as the ancient Egyptians. Are they Assyrian in the sense that they have what seems to be an unbroken line speaking Syriac since Jesus's time? Yes. In the sense that they draw a straight line Tiglath-Pileser? No. Well, sort of. When the Assyrians adopted the Syriac language, before Christianity, the entire region was Syriac/Aramaic speaking, since that was the lingua franca of the Neo-Assyrian Empire, Neo-Babylonian Empire, and the west part of the Achaemenid Empire. While most of their neighbors adopted Greek and then Arabic or Turkish, the Syrians just kept using the language. In that sense, they really are directly connected to the Neo-Assyrian which spread the language to region in the first place. Add to that some still live in the area that was the heartland of the Neo-Assyrian Empire (the Neo-Assyrian capital was Ninevah, and any proposed Assyrian Autonomous region in Iraq would include some land from the Nīnawā province), and they have as least a good a claim to being "really Assyrian" as the Greeks do of being connected to Ancient Greece or the Arabs do of being connected to the Arab Conquerors at the time of Mohammed or the Turks do of being connected to the original Turkish nomads who migrated from Central Asia and defeated the Byzantine Empire.