Now I'm not an expert on the history of Cyprus, but as far as I know before the 'Greek culture' had even become the 'greek culture' the Cypriots in Cyprus had come up with a language, the Cypro-Minoan script, because of the close relationship with the Minoans of Crete (A group of people incidental that are thought of as being ancient Greek).
http://links.jstor.org/sici?sici=0002-9 ... size=LARGE
Then the Archeans set up a few city's and then there was a large influx of Mycenaeans...But the question is, how do you know what is Greek and what isn't? Well the language for one, but there's some evidence that the Greek Alphabet was fist written in Cyprus and then transported the rest of the Greek World...so where does that put Cyprus?
Book Description
In this book, Woodard examines the origin of the Greek alphabet. Diverging from previous accounts, he places the advent of the alphabet at a point within an unbroken continuum of Greek literacy beginning in the Mycenean era. Woodard argues that the creators of the Greek alphabet, or, more accurately, the adapters of the Phoenician consonantal script, were scribes accustomed to writing Greek with the syllabic script of Cyprus. Certain characteristic features of Cypriot script, features which arose from the idiosyncratic Cypriot strategy for representing consonant sequences and from the intersection of this strategy with elements of Cypriot Greek phonology, were transferred to the new alphabetic script. Proposing this Cypriot origin of the alphabet at the hands of previously literate adapters clears up various problems of the alphabet, such as the Greek use of the Phoenician sibilant letters. The alphabet, though rejected by the post-Bronze Age "Mycenaen" culture of Cyprus, was exported west to the Aegean, where it gained a foothold among a then illiterate Greek people emerging from the Dark Age.
http://www.amazon.com/exec/obidos/ISBN= ... languageA/
I believe that Cyprus does have it's own unique culture (I also think places like Crete can also boast that), but there is a certain amount of Greek heritage that has become infused into the Island and no one can deny that, we just have to accept and embrace our similarities and differences and move on...