You could also cover the grass with mulch (wood chips, compost, coconut hulls, grape seeds, pine needles, straw-- lots of options! or the lasagna method of alternating layers of cardboard, compost, wood chips, etc.) But when I say cover, I mean really deep. more than 6" deep. a foot to 16" would be ideal! Not only will this smother the grass, it will compost it, and whatever you end up using to cover will start to break down at the bottom on the soil surface and will do the double duty job of also preparing your soil for planting. It will add beneficial microbes, nutrients, and soften the soil. It will also help retain moisture, prevent new weed seeds from germinating since they will now be that much further from the sun's rays, and prepare your bed to produce the best vegies with minimal work and no extra watering! You just have to pull back the mulch when it is time to sow your seeds.
You can generally get free rough cut woods chips from your local arborists. Give them a call. It usually saves them money to not have to bring them to the dump. A lot of them have 10 cu yd dump trucks and only want to bring that much at a time. These are not "pretty" chips that you would buy at a landscape supply shop-- they will come from a variety of trees, and you never know what you will get. Sometimes they have leaves mixed in.
10 cu yds sounds like a lot, but if you are mulching 12" deep, this much mulch will really only cover about 270 sq ft (a bed 27' long x 10' wide, or a square approx 16' x 16') for one example