As you might imagine, finding true repositories of full public domain music is difficult. I couldn't find any myself. Therefore, the "Libre" sub-category will be omitted here. Attribution and Mixed are the only categories currently.
All the music here is written and performed by one person, but you'll find varied genres from Folk to Dubstep here, and should be able to find the right piece to fit the mood you're going for.
This is my favorite on the list. Creators can set the license they use for their works, but they are all under Creative Commons, and the site requires that all works at least allow use with attribution. Some may not be used in commercial projects.
- Sample Swap
This one also has various licenses, so be sure you check the license before using these in your project. This is a repository of electronic music and loops for creating electronic music. It may not be as varied as others, but it's a great resource if electronic is what you need.
- Free Music Archive
Free Music Archive has music from various genres submitted by many varied arists. The licenses range from public domain to very restrictive, so check the license before you use the music, and be sure to attribute if needed
- The Internet Archive
I'm putting this one last, only because it's a slog to get through at times. The Internet Archive's goal is to archive everything they can, and as such you're sometimes looking for a diamond in the rough. The licenses are also more varied than the previous entries; while you'll find some public domain works, you'll also find full commercial copyrighted material, so be very careful about checking the license.