Yep. Something like that. The problem that the SBC won't be cheap and you'll need one of them per room along with an WiFi adapter. That's not even including the omni directional mike you'll need.

Barix make something called the Instreamer which is basically a network connected MP3 encoder. If you bolt this to an WiFi AP or ethernet converter then you'll probably end up with a cheaper solution and no need for embedded software development.