There were military issued revolving cylinder mechanisms on rifles.

http://en.wikipedia.org/wiki/Colt_revolving_rifle

There's one example used in the Civil War- over 4 thousand were purchased for the Civil War, and it was used at some level by both sides.

The issue was basically cost and reliability. The carbine had a reputation for discharging all it's rounds at the same time, and when it didn't, it still took a long time to reload. Furthermore there were concerns that the increased rate of fire simply meant soldiers wasted more ammunition.

The Colt revolving rifle, as has been mentioned was used in the Civil War in limited numbers. There were several issues with them though.

The first being the issue of chain fire, or all the rounds going off in rapid order in the cylinder. This is caused by the bullet not fully sealing the mouth of the chamber, which can allow sparks from firing to enter the chamber and discharge it. The solution is well lubed bullets or tighter chambers, or smearing grease or the use of lubed felt wads over the chamber mouth.

Next is the matter of spitting lead back onto the user. In a revolver, the bullet jumps a gap between the cylinder and the barrel, and enters a forcing cone that compresses the bullet as it enters the barrel. In some cases, the bullet can shear lead and spray it out the cylinder gap. Not an issue with a handgun much, but with a rifle, the supporting hand on the barrel gets splattered with bits of lead. Unpleasant to say the least.

In the end, they were expensive, five to six times longer to reload than other black powder rifles, and generally a PITA

A couple of reasons why. Both below are correct, so a bit of clarification. I am a blackpowder shooter and have a 1858 New Army revolver. Revolvers could not be loaded with as much powder as a rifle, and the reloading did take longer than a rifle. Many revolver users in the Civil War had extra cylinders loaded to make reloading in combat faster. The issue with 'all' the cylinders going off at the same time was that the percussion caps on the cylinders have an annoying habit of falling off when moved/carried/jostled. When a cap falls off the nipple, the action of firing a cylinder off can result in the cylinder with the fallen off cap firing as well. When this happens, your gun, at best, is destroyed, at worst, you are injured. Revolver rifles were also much more expensive and required more maintenance

A revolving cylinder was both slower to teach new recruits to use, and slower to reload than stripper clips and magazines which are still dominant to this day. While revolvers did see limited adoption in military rifles, clips/magazines/belts were close behind technologically and only gave the cylinder a very limited window to be adopted before being replaced.

Even those took a long while to be adopted as the predominant theory of the day was that allowing soldiers to load/carry more ammunition meant they would fire more and would be less accurate. Interestingly enough as the combat load for soldiers increased, the marksmanship requirements by the military began to diminish.

Not quite a revolver mechanism (rotary magazine with reciprocating barrel) but take a look at the 1941 Johnson. It saw some service with Marines during WW2, as well as being adopted by Chile and the Dutch for a short while.