Water the soil around the stumps really well, then get a straight rock bar (basically a long, heavy steel rod, can get them at the box stores) and start stabbing it under the surface roots. Once you get it under a root, give the rod a good push upward - the leverage should pull the root out. Walk around the tree and do this with all the surface roots. Then get the rod under the entire root ball and see if you can lever out the remaining roots.
This beats trying to hack the thing out, and it works in rocky areas where you can't shovel easily.