They spoke a type of Greek known as 'Koine Greek'. Koine is a little different to creole or pidgin languages in a rather interesting way - wheras the creole that emerged from centers of migration was a result of the interaction of different languages, Koine was when different regions spoke a version of the same language. (Alexandrian or Koine Greek was the lingua franca of the eastern mediterranean) So while there may have been a dialect in Alexandria, in mainland Greece, in Anatolia etc, traders would have spoken Koine among each other which was a sort of 'common dialect'. I'm no expert btw, just a guy fascinated with languages and their dialects.

Source: Andriotis, Nikolaos P. History of the Greek Language