Most people buy the phone cards that are usable in some of the phones some of the time if you don't mind waiting in line. Or you can call from your hotel for five times the normal rate. Or you can ask in every store, restaurant, cafe or newsstand if you can make a call. This year I found the answer to my problems. I bought a cellular phone.
Incoming calls are free.
They will also set up your answering service. They deliver your phone to your hotel, to the airport or anywhere you like, charged and ready to use. The very conveniently located Museum Internet Cafe at 46 Patision street, is an actual cafe. It is located right next to the National Archeological Museum. There are also internet cafes in Mykonos , Santorini, Lesvos and probably every other island by now.