(Star Wars: Dark Forces)
Back when I started playing games, there was no such thing as regenerating health. These games would either have you find health packs, get healed by medics, or sometimes you'd only recover health between missions. Nearly all of the shooting games coming out these days have automatically regenerating health. I'm not saying that regenerating health is inherently bad, and I'm not going to argue whether games should be realistic or not, but I wouldn't mind a bit more variety.
I guess this all started with Halo. Halo: Combat Evolved had a fairly neat health system. Your shields regenerated automatically, but if you lost health you'd have to pick up health packs. Halo 2 pushed it further with fully regenerating health (an upgraded suit of armor allowed this.) That's all fine and good, but now almost everyone's using the exact same system.
Call of Duty even dropped its health packs and started with regenerating health in COD2, and the chain reaction continued until even Rainbow Six Vegas did the same. I guess I'm not the only one tired of this though, because Halo 3: ODST and Halo: Reach are going back to Halo 1's health system.
Every now and then we'll get a unique system, like Resistance: Fall of Man's health system. You'd have 4 quarters in your health bar. As long as you're not injured below the current quarter of your health bar, you recover fully. However, once your health is dropped below each quarter, your health doesn't regenerate fully. In order to recover full health, you have to find a health pack. As much as I hated Resistance, I found this health system fairly neat. Of course, they had to drop this for the more overused "regenerate all your health" system in Resistance 2, thus removing the only thing I liked about the game.
We'll also get games like Lost Planet, where your health will regenerate as long as you have enough heat energy left, but without it your health will slowly drop (Lost Planet takes place on a very cold planet.) In order to collect more heat enemies, you have to kill enemies, cause explosions, or scavenger it. Metal Gear Solid 3 and 4 also have a neat system where you have a health bar and a hunger/stress bar. The better off your hunger or stress bar is, the faster your health regenerates. If you're in the heat of battle though, it's possible you'll have to use health packs to recover health since you won't regenerate quick enough. Perfect Dark Zero, as disappointing as it was, has partial health regeneration; your maximum health will slowly deteriorate, but otherwise it regenerates if you avoid being shot long enough. I'm sure there are other good examples, but I can't think of any at the moment. For the most part everyone uses the exact same health regeneration style and it's quickly wearing thin.
(Metal Gear Solid 4)
I wouldn't be bothered by this so much if there was more variety. If a shooter's supposed to be fast paced and intense, regenerating health is fine. If it's supposed to be slower and more tactical or involves exploration, I'd prefer health packs. Both FEAR games require health packs to recover health, and even lets you carry a limited number of them so that you can use them strategically - and I liked them all the more for it. Left 4 Dead uses the traditional health pack system, and even makes that unique because it takes time to apply the med-pack (you can't move or shoot while healing yourself or somebody else.)
(Left 4 Dead)
Overall, health regeneration isn't inherently a problem, but when nearly every developer uses the exact same system it becomes tedious fast. Don't necessarily take health regeneration away, but try to do something unique with it or use the traditional health pack system every now and then.