Historian Robert Carver writes in The Illinois Leader:
At the Constitutional Convention, the Framers looked to the examples of antiquity, the Greeks and the Romans and not to the Ten Commandments. They were a pragmatic lot, and they were not interested in being bound by their religious heritage, despite today's claims to the contrary. Rather, they were searching for virtually any idea, from virtually any source, that would work to create a better government than the failure produced by the Articles of Confederation.
Those Framers who were well educated had studied antiquity and the classics in depth, unlike the vast majority of Americans today, even those who are college educated. Thus, they were perfectly comfortable borrowing and adapting whatever suited their purposes. It would be a huge overstatement to say that they felt themselves constrained by the four corners of the Bible in finding the right government, or setting up the ultimate law that would rule the U.S.
The sources that influenced the Framers ranged from Greek and Roman law, to John Locke, to Scottish Common Sense philosophers, to Grotius. The influence of the Common Sensists was quite evident in the Framers' strong belief in the power of reason, not revelation or Biblical passages, to determine government. They were also influenced by the dominant religion of the time, Calvinism, in the sense that their worldview was rooted in distrust of any human who holds power. And this list is only a beginning.
As Carver points out, there is something strange about people asking for the government to support and promote their religion. First, the right to "religious freedom" cannot possibly include the right to have the government help you convert others to your religion. Second, if your religion is so weak that neither you nor your god can manage to keep it going, why do you suppose that the government should get involved? If the government is all that keeps a religion going, how does it continue to qualify as a religion rather than just another failed government program?