If you don't mind using chemicals, you can paint them with Roundup or brushwood killer.

Cherries root pretty close to the surface, sending out long woody roots all around - these may then start producing shoots, particularly if the main tree was cut down. Your only recourse is to locate the woody parts, drill into them to make holes about quarter of an inch deep, and apply SBK, cover with something and leave. Treating the roots in this way, if you do it properly, will kill off those woody roots and prevent regrowth.

As you have already been told, cherries root close to the surface. The only problem with trying to kill the existing roots off with chemicals is that the dead roots will come back to haunt you by creating toadstools and fairy rings in your lawn. Far better is to lift parts of your lawn and pull out the remaining roots by hand. Sorry!