Venice ruled Crete from 1211 to 1669, with slavery an important part of a complicated colonial social structure that privileged "Latins" even while, after a couple generations, basically everyone on the island had both "Latin" and "Greek" ancestry.

Through the fourteenth century, most slaves were brought to the island from Greek Europe: Anatolia (this may have included Ottoman territory, I'm not sure, but does seem to have been predominantly Christian), Greece itself, and other Greek-speaking lands around the Aegean. In the late Middle Ages, with various city-states fighting for the lucrative Black Sea trade, the incoming slave population shifted to peoples from that region: Rus/Russians, Circassians, Bulgars, and Turks.

This had mostly to do with where the navies were already active rather than any shift in "acceptability" of kinds of slaves. Medieval enslavement around the Mediterranean often, but not always, relates to general raiding activity, increased in times of war, and an inability/unwillingness of a government to ransom captives. Or, in a few cases, rebellions of a native people living under a conquest government.

Sally McKee illustrates a really interesting split between attitudes towards slavery and ethnicity in Italy itself versus in Crete and other colonies of Italian city-states. In Italy, which was (of course) dominated by "Latins", nobody seems to have been concerned with the 'ethnic' ancestry of slaves. What mattered was their low social status, bequeathed by their mother. (McKee's article here is about children of one free and one slave parent, usually a slave woman and her owner or someone her owner pimped her to). In Crete, it mattered tremendously that slaves not be Western European. They could be anything else, but not Latin. (So, per McKee, the child of a Latin father and a slave mother, after the 14th century, was generally declared free, because Latins can't be slaves.)