The Free Music Archive has all kinds of music free of the burdensome commercial licensing... most commonly the Free Music Archive songs come under Creative Commons with built-in terms of use right on the website, for non-commercial radio stations.
Easier Licensing
Having just spent an hour reviewing the conditions of Creative Commons (CC) Licensing, which apply to the Free Music Archive linked above, here are some take-aways:
CC licenses are most useful for non-commercial not-for-profit use, but terms may vary.
For example, if Attribution is required, that means the user must attribute the music to its creator, which can be done by voice on a radio broadcast, or by text in the "Now Playing" tool of a stream. The music cannot be played without the attribution.
All the music on the Free Archive is not licensed the same, there are variations in the terms offered by different artists. Most artists offer a link for obtaining commercial licensing if needed.
Small stations still need a cleric to keep track of the licensing responsibilities, but in most cases no money is involved.