Here's a study I read earlier this year. It shows how running on grass leads to less plantar forces than running on concrete.
Though this result seems obvious to me, there still seems to be some debate on the issue. In the Lieberman study, he showed that runners adjust for the hardness of the surface. While this may be true, I still don't see how can this reduce forces signfiicantly.
I'm not a physics expert, but if drop a rubber ball on concrete, it bounces back up. If you drop it on dirt, it generally doesn't. So obviously the natural ground is absorbing more force than concrete. Given that your body bounces along so to speak during running, than running on concrete should lead to more forces on the body.
Also, this example shows that running on dirt should be slower than running on concrete (because more of the force is being absorbed with dirt). After all, they do run Olympic track events on hard surfaces (like Atlanta in 1996) to increase speed.