Gear driven with a chain? What exactly do you mean by that? You also say like a bike. Do you mean a sprocket and chain? Bikes don't have gears, and gears are not driven by chains, but by other gears.
Actual gears won't work for obvious reasons.
If you mean with a sproket and chain, those do stretch and loosen over time. They are also lees convientent for small 3D printers beacuse of their weight- the stepper motors would need to be much stronger, and the rollers on the chain generally have some sort of lubricant in them. Wouldn't want a gummy dust attractor like that on my desk. Chains also are less forgiving in the "side to side" plane, so manufacturing/ assembaly tolerences would need to be more strict.
If you didn't mean sproket and chain, did you mean rack and pinion? A rack and pinion wouldn't be the best because they generally have some play in the direction they move, which will quickly ruin a detailed print.