The best way to kill grass without chemicals or machinery is to cover the area with layers of newspapers, according to About.com. The newspapers break down after several months, mixing in with the ground.
Each layer of newspaper should be about 10 sheets thick, and each section should overlap the ones next to and below it. A layer of mulch about 6 inches deep keeps the newspaper from blowing away and presses the paper firmly onto the grass. After the mulch is applied, a thorough soaking with water keeps the paper in place and helps it start to break down.