I am doing the same, i just finished replacing a fronr section, this time I used the trimmer to kill the grass in the area. It took me 1 hour that otherwise would have been a few.

In January I killed off grass in my front-yard to expand my rose-beds.. I put down pieces of roofing shingles making sure to overlap, and by the end of Feb. the grass was all dead... I like the shingles because it lets the rain water drain, also the weight keeps them in place they don't blow away... I use them in my garden to kill random weeds, drop a piece on the weed and a week later it is dead... if the shingle appears too 'ugly' just drop some mulch on it... works for me... sally

If you are trying to avoid chemicals I would avoid shingles. They are petroleum based and bound to be more toxic then any chemical available to you.

The easiest way Ive found is to lay cardboard and or newspapers and cover it with at least 4 inches of leaves and grass clippings if you want to plant in it later. Cover it with mulch if that suits you. This time of year in 3 or 4 months the grass and cardboard will turn into compost.

Try and find someone replacing carpet. Use the old carpet and lay on your grass.