I think it's more likely that Greek speakers came from Balkans. On the other hand, it's very unlikely that they also came to Anatolia from Balkans because there is no steppe ancestry and no anthropological change until 2000 BC there with Indo European names in North Syria from 2500 BC.
My guess is that even if they came from Anatolia, they would have come via the steppe regions of Bulgaria to mainland Greece and the Peloponnese. Drews in his new book sees Greco-Armenian develop in the Trialeti culture, and that's the general route he outlines.