Putting mulch under the swingset??? Do these people want splinters in their kids?
Unfortunately there is nothing that will totally prevent the grass from getting destroyed. When that happens, try to plant some high traffic grass, like what is used in soccer fields. Otherwise, you can use pea gravel or ground up rubber. Either look really nice under a swing set and are safer than mulch or wood chips. Then when the kids are done with the swing set, then clean it all out and re-grow the grass. You won't want grass under the swing set anyway unless you like to use a weed-eater.