Just to get this clear - when talking about 'layers' of newspaper,
some people may think a layer is one single sheet of paper; others think a layer is a section of newspaper, or more.
Either way, I believe
@dave is correct about the cardboard being better than newspaper, in my experience.
I hadn't chimed in earlier because I went the old fashioned route, working slowly and steadily in small amounts, using a shovel to lift and turn each piece of sod up side down - each a good healthy shovel full with all the roots - then lay the overlapping cardboard, then compost and mulch. Plant a 'first year' crop directly into the top layer of organic material and wait for year two to dig any deeper. That's how my dad did it and it has worked for me. Plus I can do this alone, without overburdening DH (which in my case is dead husband!) and feel pride in my solo accomplishment. But that's just my two cent's worth.