I actually have an answer for this....

In open wheel racing, F1, IRL, Champ Car, the drivers have what are called taer-off or tear-away strips. When their visor gets dirty, they just peel away a layer of thin plastic. So why can't the UFC and Pride have canvas systems that are held to the mat by heavy velcro? After each fight, just tear it off and velcro a new one down. Then they could clean them for the next event. Enough velcro under there and they won't move a bit. Hell, they could even buckly them down if you don't like velcro.

Thoughts?