I've checked the Rough Guide to World Music and there's nothing on Cyprus. I suspect the north part listens to Turkish Music and the south part to Greek music.From Turkey I recommend Mercan Dede, Erkan Ogur and Kardes Turkuler. From Greece try Ross Daly and
Eleftheria Arvanitaki but they're both huge countries with many different styles of music.Try a sampler to start you off.