In some cases these groups were actually legally separated from the Turks. The Ottomans used something called millets (derived from the Arabic word for "nation), which were quasi-autonomous legal groups within the Empire who were able to have their own laws, collect their own taxes, and appoint their own religious leaders. A Greek would have been part of the Rum millet (the Ottomans continued using Byzantine-style characterizations of the Greeks as "Romans"), an Armenian part of the Armenian millet, a Jew part of the Jewish millet, etc. These were mostly along religious lines rather than ethnic--the Rum millet encompassed all members of the Greek Orthodox church even if they were ethnically Balkan Slavs. There was no millet for Kurds specifically: all Muslims in the empire were under the same legal system.
As a result, a Greek or Armenian would have identified as being both a subject of the Ottomans and as part of their distinct Greek/Armenian millet. I don't really know if a Greek would call themselves "Greek" until nationalism rose, since Greek identity was always so weirdly tied up with Byzantine identity (as demonstrated by the fact that their millet was called the Rum millet), but they certainly would have seen themselves as separate from the Muslim Ottoman subjects.
More reading on the millet system:
http://lostislamichistory.com/the-decline-of-the-ottoman-empire-part-3-nationalism/
From Millets to Minorities in the 19th Century Ottoman Empire
On the Kurds, you might want to check out "Five Stages in the Development of Kurdish Nationalism" by Yavuz (2001). He argues that Kurdish national identity emerged only in the late 19th century in response to centralizing pressures from the state. Right now, you can access it for free at the journal's website.