So the good news is that Title 8 of the Code of Federal Regulations does provide for deletion of a religious oath and/or substitution of a secular affirmation.
It is true that the Oath of Allegiance for Naturalized Citizens contains a standard religious oath:
"I hereby declare, on oath, that I absolutely and entirely renounce and abjure all allegiance and fidelity to any foreign prince, potentate, state, or sovereignty of whom or which I have heretofore been a subject or citizen; that I will support and defend the Constitution and laws of the United States of America against all enemies, foreign and domestic; that I will bear true faith and allegiance to the same; that I will bear arms on behalf of the United States when required by the law; that I will perform noncombatant service in the Armed Forces of the United States when required by the law; that I will perform work of national importance under civilian direction when required by the law; and that I take this obligation freely without any mental reservation or purpose of evasion; so help me God." [emphasis added]
However, Title 8, Section 1337.1(b) provides:
“When a petitioner or applicant for naturalization, by reason of religious training and belief (or individual interpretation thereof), or for other reasons of good conscience, cannot take the oath prescribed in paragraph (a) of this section with the words “on oath” and “so help me God” included, the words “and solemnly affirm” shall be substituted for the words “on oath,” the words “so help me God” shall be deleted, and the oath shall be taken in such modified form.”
The USCIS Adjudicator's Field Manual 75.1, the policy handbook for officials presiding over the naturalization process, also states:
“Some applicants who, “by reason of religious training and belief (or individual interpretation thereof), or for other reasons of good conscience” cannot take the oath with the words “on oath,” and “so help me God” included. In these cases, the words “solemnly affirm” shall be substituted for the words “on oath,” and the words “so help me God” shall be deleted. . . . This modification may be granted upon the applicant’s request. Applicants are not required to provide documentary evidence or extensive testimony to support a request for this type of modification.”
The text of 8 C.F.R. § 1337.1 and USCIS Adjudicator's Field Manual 75.1 are available on the USCIS website and come up when you search for “oath” or “citizenship oath” through the USCIS website. The procedure allowing for modification of the oath is clearly noted on other places on the USCIS website.
If you are confronted with a written religious oath, copy the pertinent guarantees from the website of the U.S. Citizenship and Immigration Services. Enclose the documentation of your rights with the signed oath, but cross out “so help me God” before you sign it. Or, request a secular version. Prior to the naturalization ceremony, contact officials and let them know you will need to take a secular affirmation. This is usually done as a group so you will have to arrange for an alternate affirmation in advance, but the law (and the Constitution) is on your side.