How do you run without tearing up the bottoms of your feet?

The key is to gradually increase the speed or duration of your barefoot experiences, not both. For some people, the starting point will involve building up from walking barefoot. For those used to being barefoot, small amounts of running can be introduced immediately.

In all cases it is prudent to observe and not go past your limits. Overdoing it will yield blisters, calf pain, and many other types of ailments. This is part of the tearing up of the bottoms of the feet that people experience -- they are simply doing too much, too soon. You wouldn't start training and week later run a marathon, why would you think you could go out and run 5 miles barefoot without ever having run barefoot before?

I've written about (and sometimes revise) the best method of getting up to speed.

As Brandon succintly points out in the comments below, which I think is worth adding here so it doesn't get lost:

Form is a very important part of the answer to this question. You want to lift your feet, instead of pushing off with them. If you run like you do in shoes, you will rip your forefeet to shreds. The idea is to lean forward, slightly to use gravity to “pull” you, so you don’t need your feet to push as much. With this lift, there is much less friction and therefor less/no damage to your feet!
I hear people write off barefoot running simply because they do too much and find themselves hurt. Apply some common sense here.

Have something to add? I'm always listening.