So I put some callers on the air live, asking if it was legal to do that. The answer is an emphatic YES.
Turns out that the RIAA didn't want that to be widely known and they reported me to my boss. But he didn't cave and they didn't push it because they know they need radio more than radio needs them.
The radio station pays licensing fees for every song broadcast, on behalf of the listeners, and each listener has the right to make a copy of it, for himself only.
Unless I am mistaken, a similar rule applies to recording TV shows on your TIVO or VCR for your own personal use.
I’ve heard that it’s illegal to copy streams from internet radio, which wouldn’t surprise me, but is bullshit, as it is no different than what you describe. The RIAA and MPAA get antsy and suit-happy whenever a new technology comes out.