In my experience there has always been an easy and relatively painless solution to all my mechanical problems. If you are a member of an automobile club and rescue service in your country, check with them before your departure if they have an agreement with the Greek automobile club. Their staff don't always speak English but are really quite helpful. Chania, Heraklion and to some extent Rethimnon have garages representing most of the major car makes.
Parts cost about the same of course.
If your car is not a model sold in Greece you might have to wait for parts though. Most Greek words pertaining to car parts are derived from the French so if you do speak French and know the words, try using them. The same as above applies to motorbikes, although parts can at times be a little more of a problem.