chickens will eat it. fence off the area and let them run in there and they'll get it all scratching. after a year move to a new spot and garden in the one where the chickens were at. no work at all other than moving the fence each year.
the other more work approaches i do when i want to reclaim the yard here from the grass is to dig a deep hole, scrape off the top layer of sod and bury it in the hole and then put a few layers of cardboard in that hole over the sod and then put about a foot of dirt on top of that. it is very rare for anything to come up through that here. the different kinds of grasses are all different in what it takes to get rid of them. some i don't bother with any cardboard layers over them because they grow from seeds and the seeds are buried deeply enough that they won't sprout. other grasses like quack grass will come back from bits of roots and running roots that grow outwards from the center so i do have to bury those pretty deep and put cardboard over them so that they can't resurface.
the surface smothering using cardboard trick and woodchips as mulch will work too, if you don't have wood chips you can put a few layers of cardboard (overlapping the seams) and then put some garden soil on top to hold the cardboard in place, but you won't be able to grow things in there until the grass is smothered (a year or two depending upon the type of grass). it's pretty rare anything i've smothered will live into the third year unless it's escaped through a crack in the cardboard layers.
repeatedly tilling and raking the bits of grass and roots up and leaving them exposed to the sun to dry out (or feed them to the chickens). it's also a lot more work than i will do here, but some areas of quack grass need more than one attempt in this mostly clay soil before i can track down all the roots. so it depends upon how much time i have as to how long it takes me to clear an area. since i don't have a tiller and do everything by hand shovel it takes a while to go through all the clumps of clay to find all the roots. in more sandy soil it is often easier to remove things.