God is mentioned a few times in the Declaration of Independence.  The "laws of nature and nature's God" gives Americans the right to independence.  This states that it's not just the Americans who desire independence, but it is the will of a logical God.  Many of the Founders (including Thomas Jefferson) took a Deist approach to God--he did not necessarily love Americans more, but he thought it proper that they should be free.  There is the famous line "men are endowed by their creator with certain inalienable rights..." and once again, this calls George III's attention to the idea that the rights that the colonists want are God-given.  There is also a passage where Jefferson refers to the "Supreme Judge" of the world to rectify the colonial claims of injustice.  While several of the signers were Deists, religious feeling, especially in the Northeast was strong and the signers of the Declaration thought that by appealing to a higher power then they would not look like annoyed subjects but rather that they had a case for independence.  

The Constitution mentions "in the year of our Lord..." but that is a common thing in many secular documents even today.  There is also the First Amendment to consider where it says that Congress shall make no law respecting religion.  This was written in light of the religious plurality of the new nation, even back then.  Religious wars had long been an issue in Europe, and the Founding Fathers did not want anything close to an American state church or religion.