So you say, "But I can play MP3 files just fine with Windows Media Player." Well, so can I and FNV still doesn't work right. That's because Media Player uses its own embedded audio decoder in preference to any installed codec.
The problem is that OGG files (voice) play fine and MP3 files (music) do not play at all. OGG files have their own separate audio decoder so they are not affected by a misbehaving MP3 audio decoder.
It has been found in Oblivion and FO3 that codec problems are usually the source of audio problems and FNV is no exception.
I tracked this down to a codec, in this case it was the file "asaudio.ax" the ArcSoft Audio Decoder. ArcSoft codecs have been found to be a problem in both Oblivion and FO3.
The solution is either to disable the codec or uninstall the offending software package. In my case I chose to uninstall "ArcSoft Media Impressions" which was installed as part of a software package supplied with an Epson scanner. The uninstall removed the codec and totally solved the no music on the radio problem.
It is also possible to disable a codec with http://www.nirsoft.net/utils/installed_codec.html (freeware) from NirSoft. This also works but I uninstalled the software package anyway.
You can determine exactly which codec may be causing the problem by using http://www.headbands.com/gspot/ (freeware). Install it, load a MP3 file, render it with the MS A/V buttons at the bottom of the window. Button 1 will display what codec is being used - the middle item. Button 2 will play the file, if you can hear it play then the codec is working. If you can't hear the file play then you may have found the problem, the middle item in the list at the bottom of the window is the codec that is in use. You can use the "InstalledCodec" application to disable that codec, then reload the codecs in GSpot and try again until you find the codec that may be the problem.