Autosomal data provides more concrete clues: https://docs.google.com/spreadsheet/...COCa89AJ#gid=0
According to this, Turks are more than 40% West Asian, while Greeks 25.7%. This could be ancient, however, there's not a huge distance between the Greek Southwest Asian and the Turkish one (6.8% and 8.9%), suggesting a more recent influence.
11% East European between the Greeks looks significant in my book. The highest percent in all Southern Europe if we consider the Balkans apart.