We know that most Greeks buried their dead. Even as far back as the Myceneans, we know of tombs created for the burying of bodies. However, in Mythology the bodies of Heroes are often burnt. While there are exceptions, it exists throughout the Illiad and other Mythological sources. Did some city states continue to cremate? Why was cremation associated with the Golden age of Heroes? Could people elect for one or the other?
The Greeks had various funerary customs, which in most cases depended on local customs and on the social status of the deceased person. Cremation was fairly common, but that's not to say that burial wasn't. Cremation was known to the Greeks from at least the Homeric era. In Mycenaean times, the most common practice was to cremate (or fumigate) and then to bury the body of the deceased, it wasn't opting for one or another, rather it was both -- supporting evidence for this comes from Archaeological discoveries of jars found at the Temple of Hera at Olympia.
Post Mycenaean times saw an increase in cremation practices, that said, it wasn't really until the Archaic period that cremation became popular, being practiced alongside inhumation -- evidence for this comes from the Phoinike necropolis. From the late Classical period cremation declined in popularity, it's speculated that was due to cost, but the cost itself remains unknown. Support for the decline of cremation continuing into the early Hellenistic period due to cost argument comes in part from the royal tombs in Vergina, it's speculated that at that time cremation, was reserved only for royalty and higher aristocracy. When you move into Roman Greece, you see further decline in the practice of cremation until by the 2nd century when it was replaced completely with inhumation.
Sources: