When the Founders wrote the nation's Constitution, they specified that "no religious test shall ever be required as a qualification to any office or public trust under the United States." (Article 6, section 3) This provision was radical in its day-- giving equal citizenship to believers and non-believers alike. They wanted to ensure that no single religion could make the claim of being the official, national religion, such as England had. Nowhere in the Constitution does it mention religion, except in exclusionary terms. The words "Jesus Christ, Christianity, Bible, and God" are never mentioned in the Constitution-- not once.
The Declaration of Independence gives us important insight into the opinions of the Founding Fathers. Thomas Jefferson wrote that the power of the government is derived from the governed. Up until that time, it was claimed that kings ruled nations by the authority of God. The Declaration was a radical departure from the idea of divine authority.
The 1796 treaty with Tripoli states that the United States was "in no sense founded on the Christian religion". This was not an idle statement, meant to satisfy muslims-- they believed it and meant it. This treaty was written under the presidency of George Washington and signed under the presidency of John Adams.