(They spoke Aramaic long before they became Christian, of course.)
The people in question call themselves Assyrians at the present day; there are some Akkadian words preserved in their Aramaic language even now, although Akkadian itself probably died out in the earlier part of the first millennium BC.
The name "Syriac" is itself from a worn-down version of the same name; it was once used pretty much as the equivalent of "Aramaic" but is now generallly used to describe only one particular version of Aramaic which was a major literary language of Western Asia in early Christian times, and is still used as a liturgical language by Nestorian Christians as far afield as India. The script is used to write several modern Aramaic languages spoken by Christians.
These ancient communities have suffered greatly in the Middle East wars of recent times, and a huge proportion have left as refugees.