Not sure what you’re asking exactly, but the earliest ‘civilization’ in that region was probably Semitic, or possibly Indo-European. I’m not sure I’ve ever heard of other alternatives.
The Akkadians spoke a Semitic language. Not sure what the local language would have been before the Kurds arrived. The Kurds seem to be in the area by 400 B.C. when Xenophon describes the march of the Ten Thousand through what is now northern Iraq and eastern Turkey -- he calls them Kardouchoi (not to be confused with Kardashians).
There were probably a number of languages which disappeared without ever being recorded in writing.