I did some research on bikes a few years ago and came across the "chainless bicycle". I had forgotten what I read about them so I just did a quick search on "bikes with no chain" and found they are also called "chainless bikes" or "drive shaft bicycles".
The only bike manufacturer I found making them was Dynamic and their bikes are fairly expensive ($599+). They also make chain bicycles too.
I read on Wikipedia that drive shafts are a little less efficient than chains - 95% versus 98%. I'm fine with this - I've always found chains to be a bit unreliable.
