There are better alternatives. For example, use newspaper, two or three layers thick. This will prevent grass from growing through and the earthworms love it. You can also use brown paper bags the same way. Some people recommend landscape fabric, but I do not like for the fabric or plastic to show through the soil. It looks tacky. Paper will compost and disappear and the grass will have already died at that point.

Last year I didn't dig up all the grass and we used landscaper fabric and the grass grew up through it into the bed. This year we dug up all the grass for the two new boxes but I want to avoid having to dig the grass out every year. I've used boxes and paper for my flower beds under mulch and it doesn't last long so I'm not sure that is going to be a very long solution either. If I could just figure out why the grass grows where I don't want it to and not where I do, I'd be a happy girl lol!!

Are you dealing with Bermuda grass? If so, a combination of solar sterilizing the existing soil under the bed (or digging it all and running through a coarse screen to catch all the roots and runners) and burying a solid barrier against the roods and runners is the only option to constantly staying on it without using a herbicide just around the bed to constantly fight it back.

By the time the paper rots, any other grasses and weeds tend to have been smothered and their seeds rotted to the point of non-viability. Grass seeds can blow in and sprout; those you have to pluck out as they show up.