one of the issues GIG has raised is the blood relationship with Greeks and ancestral rights: well here you go:
Ancestral mtDnA - ie the female line:ftp://ftp.cordis.europa.eu/pub/inco2/docs/coe_3rd_agm_annex_3_aphrodite.pdf had been on Cyprus long before there were Greeks.
The common theory is that the Eastern med was mostly repopulated following LGM in a eries of waves in Neolithic times who mostly came out of the fertile Cresecent and/or SE Anatolia. They were of haplogroup J and E, later waves came of the Russian/Ukranian Steps who were aparent;y of haplogroup R1
Now The article shows the majority of females in Cyprus can trace their unbroken female ancestry back to neolithic times when the earliest permanant settlers arrived. That was from 6000BC to 3200BC or so. They could not have been Greek as there was no Greek or Hellenic culture until about 2000 BC - 1600 (in the Bronze age) when the Achaeans one of the four Hellenic tribes, entered what became Greece-. They founded the first Greek Civilisation, the Mycenaean Civilisation, in about 1600BC, possibly under the influence of meeting the Minoans, from whom they borrowed writing, by adopting Linear A to be Linear B.
This was at the point that Greeks can first be identified anywhere as Greeks. The Older Minoan civilisation cannot be describes a Greek - their language was not thought to be Greek nor even probbaly even a proto Greek, but they were conqured by Greeks in about 1600 - 1400 BC .
A few Myceneans (who were Greek) came to Cyprus starting in about 1400BC for trade and it is thought there were some later larger groups who came along in 1100-1000BC when pushed out of mainlaned Greece by later Hellenic Tribes, such as the Dorians.
Now it follows that if the majority of the ancestors of the majorority of Cypriots arrived ever before there were Greeks, defined as Greek Speaking/Culture, then the basic claims of Ancestral links to Greeks are weak Greekness must be a charicturistic which Cypriots aquired, I suspect probably as a result of a process of elitist cultural ascendency following two conquest events, the first in about 11000-1000 BC tevidenced by what happned in the Pafos region and Enkomi, the second in about 320 BC when all traces of non-greek culture were supressed by the Ptolomies.
Storioes of peaceful arrival and voluntary adoption by the natives of Greek language and Culture are probably ancient propoganda designed to cover up that one had a powerful cohesive group who were able to maintain their cultural identity when normally one would expect them to be assimilated by the locals. The chances are the average peasant kept his ancestral tungue for sombut he was probba;ly illiterate and few wriytten records would exist: the deeds however of the leaders would be carved in stone in their langauge.
The general drift of the gentic evidence tends however make the theories of Predominant Mycenaean descent as one of the foundations of the hellenism of the Cypriots ( as GIG keeps mentioning) probably increasingly untenable.