RoundUp and lots of it. I've used that successfuly to kill stumps. Pour it onto the cut end and keep soaking every few days. Soak the suckers.
It eventually gets the idea.

Actually, Roundup poured on non-freshly-cut wood should not affect the stump. Roundup enters a plant's vascular system through actively growing foliage or freshly cut live green tissue.

I would use a paintbrush or sponge to apply concentrated (not the thick super-concentrated stuff, just the normal stuff that I think is like 18% glyphosate) Roundup or generic glyphosate to the fresh cuts after chopping off the suckers. It may take a few doses to kill the entire root system, but it will be effective. Don't let the sprouts get too big in between doses because the suckers will re-energize the roots as they grow.

Another way to tackle this problem is to remove (mow off or cut) the sprouts promptly each time they appear (maybe as often as once a week). This isn't as fast as the chemical solution, but all stumps will eventually give up if they are constantly being cut back. It takes reserved energy to grow new root sprouts and the stump/root system only has so much of this reserved energy before it will have to have foliage to replenish its supply.

It depends too on how big the stumps are. You can rub off the new sprouts with your thumb when they're an inch long, again you'll have to do that once a week or so.
You can turn a metal bucket down over a stump if it will fit and pile dirt around the open end, blocking out the light. This will kill the stump eventually, but you have to leave it all summer. You could get a goat and leave them in the area with the stumps to nibble the new growth.