Try cutting a piece without music, then find some music you like and make it fit by either slowing down some shots or lengthening or trimming to fit the music track. As the others have said, a few cuts on the beat at the beginning and a key transition points are good, but not to be used every time.
If you have a track with the occasional drum fill or guitar riff, try cutting to that, with natural transitions in between.
I know you said these are quick turnaround, but you could do a few easy effects to hit on the occasional beat, such as freeze and isolate a person doing something, like jumping over a log or whatever.
Sounds like fun.
Mike