The Greeks didn't have a priestly class with any real authority like the Vedic Indians, ancient Israelites, Egyptians, etc. It seems like the Mycenaeans did, but at some point during the 400 year dark age after the Bronze Age Collapse, they were displaced for unknown reasons. Classical Greece technically had priests, but they were completely subject the polis and and didn't do a whole lot other than maintain the temples, etc. You didn't have to be a priest to perform a sacrifice like in many similar societies. Any male citizen could do it. The lack of a priestly class in Greece to control the society's behavior through established dogma, etc., freed poets and (later) philosophers up for religious and cosmological speculation.