A dog?

rose maybe work.

Anything they can't scratch up. Maybe loose pieces of flagstone. Those can be taken up as needed to accommodate growing plants, or once the cats have lost interest, which they might. Lay them with narrow enough gaps that a cat's paw can't get access.

But other creatures scratch and dig too, so I think rearranging is just part of having mulch. You might look into some ground cover plants that would grow to cover the area instead. Something creeping like the rug junipers or something herbaceous that spreads via roots.

You might also choose plants that will cover a good patch beyond their stems, like hostas or mature peony clumps, and plant them close enough to touch.

KarinL

You might look at some of the suggestions here:

http://landscaping.about.com/od/pestcontrol/a/cat_repellents.htm

http://forums.gardenweb.com/forums/load/pests/msg0807514611805.html (see the 4th post; it also mentions some things not to use)

Or other sites from this Google search:
http://www.google.com/#hl=en&pwst=1&sa=X&ei=QXMfTvGcN-bX0QH1tKjdAw&ved=0CDcQvwUoAQ&q=cat+repellent+garden&spell=1&bav=on.2,or.r_gc.r_pw.&fp=d97c07d6cfed538e&biw=1372&bih=813

I think most cats hate the smell of citrus, so maybe if you grew plants that had that kind of smell it would help deter them.

Are these your cats or neighborhood cats?

Lay thorny twigs (roses, mesquite, whatever) across the mulch in a criss-criss pattern. They hate walking on thorns.

For something a bit more gardener friendly, lay lattice across the mulched areas. They don't have room to dig.