Bobby R says:

Concrete has small groves that trap fine dirt and dust. The dust is released when your soles sink in between the groves and make contact with it. Concrete close to roadways has oil and rubber particles from cars too. This combined with the dust makes for very black, dirty soles. The oil makes it adhere more to skin.
My wife, kids and I were out barefoot last weekend. We were just walking down the sidewalk on our street saying hello to a neighbor. By the time we walked back home, the bottoms of all of our feet were covered with black, dirty soot. I know what you mean!

gar_fana says:

If you’re walking barefoot on concrete, you have much tougher feet than I have.
To answer your question, the grass tends to clean the feet, wood probably doesn’t have much dirt on it, therefore both are less dirty than the concrete is.