I think audio recording is one of the few things you're allowed to do in the background. I didn't do anything special. Did they give you any specific reasons?
On the other hand I assume that you don't need to keep long audio samples to run your alarm-recognition algorithm and just discard the audio almost right away.
That's just a wild guess of course, it would just seem reasonable to me if the Apple reviewers saw it that way.
I also found other people with the app being rejected for the same reason after asking about it in the apple dev forum.