Hi, as others have said, in before the Balkan Wars in 1912-1913 the region of Macedonia, Albania, parts of Bulgaria and Greece were still part of the Ottoman Empire (modern Turkey).
Salonika would be today's Thessaloniki, Greece. "Cashoria" sounds like it's probably Kastoria, Greece.
Based on my experience, people that immigrated to Anglophone countries tended to use "James" or "Jimmy" for the original name Dimitrios or Dimitris (Mitre in vernacular Macedonian). Thomas in Greek would either be Thomas or, less accurately, Athanasios...possibly even Anastasios. As surnames these could take a number of different forms, but usually the genitive case of the male name is a good pattern to start with if you don't have other information. My thoughts would be that possible surnames might be Thomas, Thomaidis, Thomakis, Athanasiou, Athanasiadis, Anastasiou, Anastasiadis.
The other tricky thing is how your ancestor identified ethnically, and what language they spoke. The name they used (and may have passed down to you in their oral history) might not be the name the Ottoman Empire listed them under, and not the name the Greek government would recognize today. It's also possible that they or their relatives had several different translations of their names. (My grandfather and his sister had their surnames recorded differently by authorities in Greek, even though they were of course the same surname in Macedonian).
I hope that's helpful info!