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The founding fathers were at considerable pains to avoid any establishment of religion.

The Framers of the Constitution seemed to go out of their way to assure that result:

Article I, Section. 3
Clause 6: The Senate shall have the sole Power to try all Impeachments. When sitting for that Purpose, they shall be on Oath or Affirmation.
Article II, Section. 1
Clause 8: Before he enter on the Execution of his Office, he shall take the following Oath or Affirmation:--"I do solemnly swear (or affirm) that I will faithfully execute the Office of President of the United States, and will to the best of my Ability, preserve, protect and defend the Constitution of the United States."
Article VI
Clause 3: The Senators and Representatives before mentioned, and the Members of the several State Legislatures, and all executive and judicial Officers, both of the United States and of the several States, shall be bound by Oath or Affirmation, to support this Constitution; but no religious Test shall ever be required as a Qualification to any Office or public Trust under the United States.

The First Amendment is icing on the cake:

Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof ...

For those who may not understand the expression "oath or affirmation" it's explained here: Affirmation (from Wikipedia):

In law, an affirmation is a solemn declaration allowed to those who conscientiously object to taking an oath. An affirmation has exactly the same legal effect as an oath, but is usually taken to avoid the religious implications of an oath. The Constitution of the United States makes four references to an "oath or affirmation": In Article I, Senators must take a special oath or affirmation for the purpose of sitting as the tribunal for impeachment; in Article II, the president is required to take a specified oath or affirmation before entering office (see oath of office); in Article VI, all state and federal officials must take an oath or affirmation to support the U.S. Constitution; and in Amendment IV, all search warrants must be supported by evidence given under oath or affirmation.