Revolver rifles did exist, the Colt 1855 being the most produced variant, but not in large numbers and for good reason. They were rather expensive as revolvers require tight tolerances and this need for precision only increases with larger cartridges. This meant that practically all revolver rifles often utilized under powered loads compared to their breech loading and muzzle loading brethren. For this reason, they are more appropriately called carbines as they are essentially large powerful pistols.

Another problem is that, once you expend your first cylinder, you have to reload them in essentially the same manner as a muzzle loader: insert powder, pack powder, insert round, pack round, and place cap. From a military perspective, it means that after the first 30 seconds of battle you are now on an equal footing with muzzle loaders.

The most glaring problem though is with all the loose powder flying around you open yourself up to chain fires. This is when the gas escaping from the primary chamber causes all the other chambers to fire. This isn't a problem with a handgun because it just means your six shooter became a surprise shotgun; however, in rifles it means you lose your off hand. Due to chain firing the few US army units that did utilize revolving carbines instituted policies that made them worthless (e.g. Only loading one chamber or requiring both hands placed behind the cylinder).

Once metal cartridges became more widespread lever action(which is actually adopted from early revolvers) and bolt actions became prominent, but that's a different topic.

Disclaimer: I'm simply a firearms enthusiast and not a historian by any means.

Edit: Grammar and spelling

You might consider the story of the Puckle gun -warning, pdf!- devised by James Puckle in 1718. It was a flintlock revolving rifle. To somewhat meet the problems of precisely mating cylinder and barrel bores ( done on the Colt and other revolvers with a cylinder stop) Puckle made the base of the barrel chamfered, the end of each of the chambers of the cylinder turned conical to match, and a simple screw pressed the chamber into the barrel, providing a simple seal that was, probably, effective enough. It was never found to be practical- even though it offered the thrilling chance to shoot square projectiles at infidels, and decent round balls at Christians. As others have posted, the rapid-fire of a revolver depends on how much time you've had beforehand to load the cylinder, and whether there are enough cylinders to last a battle.