We recently had a very large cottonwood removed from our yard. I have been finding twigs that are sprouting roots in the dirt and mulch left behind after the stump was ground down. I hope I don't have to worry about the darn tree growing back, since the twigs seem to be very robust. Thoughts?

Karen, Cottonwoods are very tough trees. If the stump was ground down any suckering from the roots should be minimal. their stems root easily as you have found out. Removing them should alleviate that problem.