You can use music in your podcast if:
It’s the public domain (and be absolutely sure, since penalties are harsh) or
It has a Creative Commons (CC) license that’s compatible with your use, or
That you otherwise have an explicit license(s) or permission to use
Because of the way podcasts and music rights work — podcasts requires the same licenses one would need to reproduce and distribute a CD — in practical terms this means you're limited to:
Public domain music
CC-licensed music (with a license compatible with your use)
Confusion around "fair use" is a constant. Here are resources to help you understand if your use of copyrighted materials may fall under U.S. copyright law "fair use" exemptions.
Fair use isn’t a right, but a legal defense.