I've read a couple of books on ancient Greeks lately. The first one was a book of Plutarch on the oracles of the Pythia of Delphi. Plutarch, best known as an historian for his Parallel Lives, was also for some time a priest of Apollo, so he gives there some first-hand informations.
The second book was a very interesting essay by Paul Veyne, a contemporary French archaeologist and historian, titled Les Grecs ont-ils cru à leurs mythes? (Have Greeks believed their myths?). It begins by general considerations on the ancient Greek religion, and the place of mythology herein; then it takes a more philosophical turn, in the steps of Michel Foucault, and discusses what the notion of truth means.
One of the things that can be noted about the Greek religion, and that is much apparent through those two books, is that Greeks, like many ancient civilisations, did not have a concept of "faith", which we now in the modern western world tend to consider as a common ground for religions -- probably because we're mostly only familiar with the two big monotheisms, Christianity and Islam. No faith: nobody, in the home country of philosophy, would have considered virtuous or honourable to hold a mandated belief, not meant to be discussed or to be subject to inquiry. Incidentally, nobody was scandalised by the numerous philosophers who were arguing about the trustfulness of the oracles or the existence of the gods.
However, while many Greeks simply did not believe in the gods, and even less in their adventures as told by Homer, Hesiod and all the mythography, it appears that they couldn't imagine that those myths were completely invented, and they tried to explain their existence by several theories: for example, that gods were great kings of the past, later divinised, or that the myths were hiding ancient doctrines hidden under allegories.
So what role did myths play in the Greeks' everyday lives? I think that they were part of the tradition, in the noblest sense of the term: the tradition as the foundation for a culture (and that's why they ought to be respected). The myths, the gods and the heroes were common figures, models for good or bad behaviour, a common language of stories and situations that everyone could refer to. Probably, this profusion of the mythical language, and the freedom with which it was treated, created the ideal ground for all the inventions that were made in Greece: mathematics, geometry, theatre, democracy, philosophy, history, and so on.