Ancient Greek dramas were actually presented as religious festivals, most famously the festival of Dionysus in Athens. The themes of Greek drama often had a strong religious background.
One concept that affects the plots of many ancient tragedies is the notion of "miasma" or ritual pollution. A person who offends the gods in certain ways brings ritual pollution not just upon himself but upon his city and descendants. Thus the unburied corpse in Antigone, the matricide in the Oresteia, the curse of the House of Atreus, the patricide of Oedipus etc, cause miasma and thus bring down vengeance of the gods on entire cities. The plots of the plays revolve around how to propitiate the offended deities.
Other major religious beliefs include the inescapabiliuty of fate and the ability of prophets to foretell the future. Both of these present us with a spectacle in which no matter how much the tragic hero or heroine attempts to evade his or her doom, it nevertheless is unavoidable.
Finally Greek religion operated on the principle of "do ut des" (I give that you might give) and the characters in plays are constantly trying to solicit the favour and avoid the disfavour of the gods.