My LG Nexus 4 (Android 4.2.2) does not play my .wma audio files. However, my generic/budget Android 4.0.3 tablet plays the same .wma files just fine.
How can I get my Android device to play .wma files?
It seems that ICS does not have native support for WMA files.
You can use an app like MX Player (or its Pro version) to play WMA files.
A bit of a workaround, but... If the music (.wma) files are part of your music library then you can sign up to Google's "Play Music" and upload all your library to the cloud (20,000 songs for free). The resulting streamed music appears to be in a format that the device can play.
The "Play Music" app still can't play .wma files directly that are stored on your device. However, the streamed music is saved/cached and plays offline.
The "Music Manager" that runs on your PC uploads all your music tracks (.wma files) to the cloud. However, it doesn't upload songs that are already in its music library. I thought this was maybe the key - that the same song already exists in a compatible format in Google's storage - but it works with any .wma file, even files I have recorded myself.
EDIT: If I download a track, that was originally a .wma file, from my cloud library on the web then it downloads a .mp3 file. This is the same for all .wma files - even files I have recorded myself, so Google must convert them.