We like to mix some compost into our soil, cover with a couple layers of newspaper then apply either half-finished compost and a mix of weed free grass clippings. This allows the compost and grass clippings mixture, mostly just the clippings, time to breakdown without too much nitrogen from the soil. Wood chips and other non-broken down materials will take nitrogen from the soil as the break down. Newspaper does, but IMO, it seems to take less.
Chopped straw works well as long as you're sure it's weed free.
You can also use landscape fabric.
We also use living mulches, such as hairy vetch. Plant them in the winter, cut them down in the spring and plant directly into them. No weeds and tons of nitrogen and great soil.