Why do you want to remove the roots? Any way you do it is gonna rip up your yard.

If you're concerned about suckering, mowing will take care of them and the roots'll be dead after one season with the tops gone. If it's an area that doesn't get mowed, find an agronomy business and buy a 1 quart bottle of Tordon RTU for $10-$15. Recut the stump and apply the Tordon to the fresh cut. It won't stand a chance.

If the roots are close to the surface, mowing may not be an option. The mower might hang up on them and bend the blade/crankshaft. Knocking them out with the Tordon, a truckload of topsoil and re seeding might be the answer.

A stump service will probably be the cheapest route of the professional services. Otherwise, you are looking at heavy machinery coming in to dig up your stumps and tear up the yard where the equipment moves around, which you will then probably want to repair.

If you don't go the natural method of killing the trees and letting them rot, then you can rent a stumpgrinder. I'd get the biggest one you can handle and move. Poplar is not tough but the bigger machines make the work so much easier. I'm guessing you'd be at ~$500-1000 for a pro stumpgrinder to do the stumps and roots, remove some of the grindings, lay down dirt, and reseed. You could probably do it on your own for ~$200 (depending on how much soil you needed), including rental, dirt and seed. It will take you a few hours though.