I came across this article, which describes four ways to remove grass in order to create or expand a garden. I admit I’ve tried versions of one through three (but without the plastic layer). Method three, also known as the lasagna method, definitely gets my vote. If you cover the grass with mulch as well as newspaper or cardboard, it won’t be unsightly and you’ll feed the soil. One tip the article doesn’t mention: mow as low as possible first.
It’s also good paired with my far less sophisticated version of Method One. (I just dig. Or hack with an ancient tool I was given but can’t name).