Answer: If you keep mowing down the suckers as soon as they appear, the roots will eventually run out of energy and will not be able to produce suckers. It's a long process, but the easiest, and the safest for your lawn. Or, since you know where the roots originate (where the stumps were removed), you can dig down, make clean cuts in each of the roots and paint each fresh cut with full strength Round-Up. It is a non-selective herbicide, so don't use it on your lawn, and keep it away from desireable plants. The herbicide should travel through each root and kill it. Wish I had an easier solution for you!
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