It does matter. This lie is perpetrated over and over.
The Turkish tribe that converted to Judaism was called khazars, the converts were called karaims. Their temples where always separate from synagogues. Khazars and Jews lived separately and, generally, did not inter-marry. Slowly, khazars as a nation disappeared. There are few links to karaims on the Internet.
http://www.google.com/search?hl=en&ie=UTF-8&oe=UTF-8&q=karaim Most of Ashkenazi Jews of today are direct descendants of ancient Hebrew tribes.