I don't recall the last game I played that had regenerating health, but this is definitely true; when all you have to do is sit around fingering your own bunghole until you get back up to health, it takes a lot of the challenge out of a game. Yahtzee's alternatives to randomly spawned health items are good, but if it hasn't been suggested yet, I want to toss in my own suggestion: Health by Self-Medication.
This isn't an unexplored concept. The best example I can give right now is either Killing Floor or LFD/LFD2 on Steam. For those of you who haven't played the former game, every player starts out with a syringe that injects some sort of painkilling/health restoring drug (maybe there is a future in medicinal marijuana, ba dum tish.) When the players find themselves injured, they can use this syringe to inject themselves with drugs that will, after a few seconds, boost your health back up. You get an unlimited supply of vials of drugs, but it takes a while before you can use them again. Also, you can use the syringe on other players for even greater effect and vice versa.
I think this kind of system is a very good one; it's not effectively giving you unlimited health in the middle of a fight because the rate enemies injure you is a lot faster than the rate the drugs heal you. Plus injecting yourself requires you to put away your weapon and pull out the syringe, so while healing you can't fight back. The best strategy is to find a lull in the battle (often provided by team mates covering you) and either taking your meds or having someone else administer them. This kind of system seems like a great way to keep health item drops to a minimum.
If you want to have at least SOME health items, I would advise going the LFD/LFD2 route and be able to carry around first aid kits. To use them you have to stop and either bandage yourself or get someone else to bandage you. Now I sometimes find it a little surprising that the entire contents of a first aid kit would be used by one person, so maybe the system could be revamped more like Call of Cuthulu: Dark Corners of The Earth, where the first aid kit can be used more than once, but still a limited number of times.
Whatever the model, I think this method of healing would really work well. It could reduce or do away with having to drop health items like confetti all over the game while still not making recovery as brain-dead easy as just staring at a wall. You'd have to pick your time to heal up, possibly have to ration your carry-on healing supplies, and it could even involve a mini-game to determine how much health you get out of the experience. I know Yahtzee said he doesn't like quick-time events very much, but he did say once he could understand them as a core part of gameplay rather than a random, no-warning-dick-move sort of element. This could be a good place to apply that.