I read a time travel story once where a present day American finds that most women in ancient Babylon think he's amazing in bed because the ancient near East culture puts no importance on pleasing women in bed, consequently most men didn't bother. Is there any evidence that's true?

I believe this question is impossible to answer. While some of the worlds cultures practice female circumcision, other cultures may be practicing the Kama Sutra - there may be satisfied women in oppressive cultures or oppressed women in liberal cultures. (EDIT for clarification- That Babylonians did not use female circumcision, just an example of cultural practices that restrict female sexual enjoyment.)

Ancient Babylon predates the Kama Sutra by ~1,500 years. In the time you are speaking of, a woman's sexual pleasure was poorly understood, but some women had satisfying relationships, while other women were raped. The diversity of human sexual experience makes this question difficult.

While trying to find a more specific answer, I came upon some Babylonian Erotica. The sexual positions don't look markedly different than any we use today. Dr. Julia Assante is quoted saying the Babylonians had “an exalted cultural view of sex as inducing an altered state of wonder.”

According to Dr. Assante, anal sex was used as a contraceptive. Oral sex was had for fun. There was prostitution.

Verdict- It probably depended on your partner.