Atheist can simply say, "I affirm..." and it is as good as "swearing on the Bible." Many faiths see "swearing an oath" as a sin...so this is what they do...instead of "swearing on the Bible".
I don't see the controversy here...both philosophies see swearing an oath as something unnecessary...simply tell the truth...it will take the complexities out of your life.
Honesty is the best policy...not just a cliche...it's what's for breakfast. Eat it up.