What do you mean no one there calls themselves Assyrians? Surely you've heard of the Assyrian Christians?
http://www.assyrianchristians.com/
This is the "Holy Apostolic Catholic Assyrian Church of the East" pejoratively called "Nestorian".
This is the address of their official website:
There is also a version of this group that is in union with the Roman Catholic Church who go by the name Chaldean. They both speak Syraic. I'd encourage you to follow the Assyrian links portion of the webpage.
One last thought ... do you have Goode's World Atlas? If so, turn to the map in the European section showing the languages of Europe and the Middle East. IIRC, Assyrian is listed and is shown in northern Iraq, southern Turkey, and northeast Syria.
Look at the cached pages here also (the actual pages seem to be down right now):
http://www.google.com/search?hl=en&ie=ISO-8859-1&q=Assyrians