This is not true my friend. I wish some Turkish poster was here to explain to you, but the truth is that a HUGE number of Eastern Roman Greeks ("Byzantines") converted to Islam and became Ottoman and then Turkish after a couple of generations.
Sometimes when I travel to Turkey and I walk the street people just talk Turkish to me as they think I am a local. Same thing happens to us, we see Turkish people and we swear to ourselves that they are Greek but ofc they aren't, they are just Turks from Turkey.
A Greek Army General called Frangos actually wrote a big book based on official documents of the Orthodox Patriarchate based in Constantinople (now Istanbul) which continued to exist as an organisation after the conquest of 1453 with the support of the Ottoman State and Sultan Mehmed Fatih.
In it, Gen. Frangos lists with incredible detail the fate of the Orthodox Dioceses in Asia Minor after the Conquest, and you can see in the official data the decline and death of Christianity in Asia Minor and with it the decline and death of Hellenism as in many cases, whole towns converted en masse to Islam on a single day, including the former priests who became the town's new Imams.
In fact, so many former Greek Romans ("Byzantines") converted to Islam (and thence to Turkism) that the Ottoman Sultans were alarmed and issued decrees to discourage Christians from converting to Islam because under Islamic and Ottoman Law, Christians were taxed significantly more than Muslims in the Ottoman Empire and the Ottoman State was worried that they would lose way too many taxes.
It is 100% true and written by one of our own Army Generals.
There were somewhat less converts to Islam in Ottoman Greece and that can be attributed to the incredibly stubborn national character of the Greeks of Greece and their national pride. Also, the Ottomans were never seriously interested in colonizing Greece with Turks and Turcomans as much as they did Anatolia. To the Ottomans, everything beyond Istanbul was "Rumeli" meaning "Roman Empire" and "Yunania" meaning Greece.
If the Ottomans were seriously interested in converting Greece to Islam they would have succeeded, but it was something they could not do b/c they feared two things, the Pope and the Italians on the West (Venice, Genova) and the Russian Empire on the other end.
Then you have cases like Bosnia and Albania, where the population converted to Islam willingly and en masse for reasons of national pride and to separate themselves from their arch-enemies, Greeks and Romans in the case of the Albanians and Serbs in the case of Bosnians.
Book by Gen. Frangos detailing the decline and extinction of Hellenism in Asia Minor:
http://www.livanis.gr/%CE%A6%CF%81%C...p-1086175.aspx