Perhaps I've mentioned this before, but an important factor in figuring in what you want to sound like is figuring out what you don't want to sound like. I recently was in a situation ( not teaching, mind you ) where I heard a bunch of bands rehearsing, and I was reminded of what I ( and maybe you ) need to focus on, because I wasn't hearing much of it.
1. Dynamic range
It's very fun to slam the drums and play aggressively but remember, differences in volume is how we create life and drama in the music.
2. Different styles
I feel it's important to work on different types of grooves, rather than just the ones we can play already or feel comfortable with. In my case, a lot of world music styles I have practiced have helped me immensely, even though I haven't played many of those type of gigs.
This is a crucial aspect of drumming, whether you play Jazz or not.
5. Music and tune learning, rather than "lick" learning