I'll readily admit I don't like the standard rapid healing rate, but I'm pretty enthusiastic about avoiding house rules. This isn't a panacea, but one solution is to focus on running episodic adventures book ended with long rests. The long rest healing happens off stage so it doesn't really disrupt verisimilitude.
Another solution is to make the rapid healing unique to the PCs for story reasons. I ran a game that began with the PCs being the sole survivors of a mad wizard's attempts to create a super soldier. The treatment imbued them with superior natural healing and would also function to explain the rapid growth in power experienced by PCs. Meanwhile, the rest of the world healed (and acquired power) at a more "realistic" rate.
Neither type of solution may work well for your game, but I thought they may be useful examples of a different way to approach the problem.
I've actually thought about doing something like this, and I might for future campaigns. This idea, or even the troll inbreeding idea that was posted earlier would satisfy the rational that I'm looking for.
I would like to have one solution to use in all my games though. The search for perfection continues.
