We know, certainly, how much the Greeks influenced the thinkers of the Revolutionary Era. The founders were perhaps among the last generations of the inheritors of the Renaissance of Greek knowledge that emerged after the "Dark Ages" and blossomed fully during the Humanism fostered by Erasmus, who taught himself Greek that he might go back to the Septuagint, or men like Thomas More who learned Greek partly to understand the Greek influence on Cicero. Erasmus was actually worried that concentrating on Greek culture too much would result in some kind of neo-pagan undercurrent to Orthodox Roman Catholicism.
Our founders, of course, we past all that--but with such a debt to the Greeks (even if it was by way of the Roman Greco-philes) is it possible that some of the founders may have been attracted to paganism? I don't mean to say that any of them actually worshiped Zeus or Mithras, but it is easy to suspect that some of the more tight-lipped Founders secretly modeled themselves after the Greek philosophers who payed lip service to the state religion, revered the mythology, and yet maintained a certain agnosticism.
One small example of this comes from James Madison. The quote I'm about to pull has been discussed here before, but in a different context.
"The finiteness of the human understanding betrays itself on all subjects, but more especially when it contemplates such as involve infinity....The infinity of time and space forces itself on our conception, a limitation of either being inconceivable; that the mind prefers at once the idea of a self-existing cause to that of an infinite series of cause and effect, which arguments, instead of avoiding the difficulty; and that it finds more facility in assenting to the self-existence of the universe, visibly destitute of those attributes, and which may be the effect of them. In this comparative facility of conception and belief, all philosophical Reasoning on the subject must terminate."
This seems to be a more eloquent echo of something the Greek pre-Socratic philosopher Xenophanes said many, many years earlier:
"The certain truth there is no man who knows, nor ever shall be, about the Gods and [other theological matters]. Yea, even if a man should chance to say something utterly right, still he himself knows it not--there is nowhere anything but guessing."
I'd like to explore the pre-Christian influences on the Founders a lot more.
Sources: The Madisons at Montpelier by Ketcham, The History of Western Philosophy by Russell, Ad Infinitum by Ostler.