Alternatives to Diffie Hellman
The Wikipedia page on "Post-quantum cryptography" provides a list of proposals for PKE resistant to quantum attacks. Quantum algorithms can solve DL in finite abelian groups (as well as a few nonabelian and infinite abelian ones), so they get very close to the spirit of the question you are asking. The learning with errors problem mentioned by one of the commenters is on that list.

I do not know of a general answer to the question of "What is the power of DL in arbitrary (that is, nonabelian) groups?".