I cut down some small trees how can I eliminate the roots w/ no diggin

Time and a lot of cussing!! Lol!! There's a product called "stump out" or "stump be gone" (this one comes with a wand:)). Seriously I think the Stump Out is supposed to rot the old stumps. Or have you thought of having them ground down by a tree company? For the roots, maybe a heavy application of Round Up (just be careful of nearby plants and you may have to apply more than once).

I am not a fan of roundup. Drill holes and pour vinegar in them. Lots and lots of vinegar. Then wait. Since we are going into the summer season, it should go quicker.

Copper nails

I used roundup on a stubborn wild grapevine. I cut into it with a saw and carefull put it only there and it killed it and none of the hostas or ferns that were growing close to it. She wants to get rid of the roots, not kill the roots. If you are wanting to plant where the trees were, just dig your holes and cut off the offending roots that are in the hole. I have done this many times without a problem. No matter how hard I try to dig up roots they seem to show up anyway, and not always just the little roots.

Here's a great way to kill tree roots or the stump without digging.....http://homeguides.sfgate.com/kill-roots-stump-tree-digging-up-44878.html