|Mr. Charisma pulling up the radishes|
As I mentioned before, I shred paper and thin cardboard* and then let it soak overnight in a bucket. In addition to adding it to the compost pile, I also tried using it as a mulch directly around some of the bigger plants, like tomatoes.
It worked very well. The wet paper matted together and stayed matted even after it dried, so I didn't have paper blowing all over the yard. But it let the rain through while keeping weeds from going the other direction. When I pulled it up, the bed beneath was crawling with the kinds of bugs you want in your soil.
Once I cleaned out the beds, I threw most of the half-rotted paper on the compost pile, while some I left on the beds to see what it will do over the winter and what it will look like in the spring.
*The thicker cardboard I'm using directly on the ground around/between these beds, as well as in a new, partially-shaded area on the north side of the house that's becoming a garden.