Well, there have been some revolving rifles, but they were not very successful. The main issue (and this was also an issue for early revolvers) was a situation called a "chain fire".
Remember that the early revolvers, such as the Colt Paterson and the Walker Colt were using percussion cap technology. Upon firing the weapon, the gunpowder would burn in the chamber and propel the bullet out of the chamber and through the barrel. The flame would then spread out from the front face of the chamber and sometimes go into the next chamber. If the bullet in the next chamber was tight fitting, this wouldn't be a problem, but sometimes users would use a less tightly fitting bullet and there would be gaps between the bullet and the chamber wall. This would sometimes cause gunpowder to leak out of the chamber. As a result, the gunpowder in the next chamber would ignite as well and shoot out its bullet from the side of the revolver, not the barrel. If the user was unlucky enough and used loosely fitting bullets in all chambers, the flame would travel from chamber to chamber and ignite each one, thereby causing extra accidental discharges.
The same situation would sometimes occur if the user used larger (looser) percussion caps on the nipple. When the percussion cap was struck, the flame would travel through the nipple into the chamber and ignite the gunpowder. With a looser percussion cap, some of the flame would travel outside the nipple and through an uncapped nipple or through the loose percussion cap of the next nipple and ignite the gunpowder there.
The result of a chain fire is multiple discharges of the other chambers of a revolving weapon. With a revolver handgun, this is not as much of an issue, besides giving the firer a surprise, because the handgun is usually held at arms length and with one hand only, which is behind the cylinder of the weapon. As the extra chambers fire, the bullets leaving them don't travel as fast because they don't pass through the barrel. They also can't injure the firer, as his hand is behind the cylinder.
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On the other hand, with a revolving rifle (such as the 1855 Colt Root model above), the user holds the rifle with two hands and one of these hands is placed in front of the cylinder. So when a chain fire happens, there's a good chance that the user could get his forearm and hand injured by the other accidentally fired bullets. As a matter of fact, the Colt Root model was used for a while during the US civil war by a unit called Berdan's Sharp Shooters. Reports from that era indicate that a number of people set their sleeves on fire from chain fires and that you could tell how long a sharpshooter was in that unit by counting his remaining fingers. The Colt Root was even nicknamed "Colonel Colt's Revolving Wheel of Misfortune". Commanders attempted to counter this by telling their soldiers to hold the rifle in such a way that both hands were behind the cylinder, or by telling them to only load one chamber at a time to prevent chain fires. The 21st Ohio Volunteer Infantry used this rifle with some success in the Battle of Chickamauga, but the problems of chain fire caused the Army board to recommend dropping this weapon altogether.
Later revolving rifles used cartridge technology, where the problem of chain fire did not occur. As a result, revolving rifles made by D.B. Wesson (of Smith & Wesson fame), Remington, Colt Buntline etc. found some use. However, the problem of escaping hot gas from the front of the cylinder burning the user's forearm still remained and this is why they didn't become popular. Another big issue was keeping the weapon clean in combat conditions. Remember that a revolver design has some gaps between the cylinder and the barrel. With a handgun, the weapon can be kept relatively clean by storing it in a holster, when not in use. On the other hand, rifles are carried without a carrying case. In muddy and dusty conditions, keeping the internal mechanisms clean is a lot of work.
However, the idea hasn't entirely vanished. Some auto-cannons, such as the Mauser MK213 and the Aden auto-cannon use the same idea, but they are aircraft guns, not firearms. Some grenade launchers use the same concept as well.