These are coming off roots below the soil - the usual culprit is a mature cherry tree nearby, because they put out long and large roots just under the ground, and over time, those roots produce this type of growth. This can also happen after such a tree has died or been cut down, unless the stump and roots have been bored out. You can expose the roots, drill into them about a quarter inch deep every foot or so, and fill the holes with SBK, neat, without spilling it anywhere else, cover with something. You won't be able to replace the grass till next year though, and this treatment may, or may not, kill the parent tree/shrub, wherever it is. Otherwise, just keeping them mowed down is a partial solution, which presumably you're already doing.

I have a similar problem with my flowering, weeping cherry but I don't want to harm the tree itself. What should I do with the sucker growth?

Nothing you can do Patfran except wrench or cut it off when it grows.