I'm a accredited Billings Ovulation Method teacher but am also Catholic. One of the things we had to agree to in becoming teachers was to follow the Church's teachings on family planning. Billings is not FAM though (no barrier methods allowed, abstaining during fertile times). IME, most NFP organizations are associated with the Catholic Church.
Just googled a bit and found
Justisse. Haven't heard of them before but seems to be a secular organization.