I strip the cellophane tape off of the cardboard before I pass it through my wife's paper shredder. It yields little rectangles about 1/4" x 3/4" of an inch, which is a good size for adding to my compost bins. I don't overload the bins with too much at one time, and I make sure the greens are in more prevalence than the browns. It works well. Usually by the time I turn the bin, all evidence of the cardboard is gone.
I also moving toward the approach practiced by Charles Dowding in the U.K. where he lays cardboard right on top of the weedy ground, tops it off with a layer of compost, and sows transplants directly into that. Apparently the worms and other decomposers like to eat cardboard!
Thats actually what i am lookin to do. I mulch about 40-50 yds of debris each year from our forest (dead trees and such) and wanted to amend the mulch. I have ust read so many different viewpoints on the use of cardboard in that way. I was looking ot add maybe 10 tons to the mix.
Do the next thing next. That's a pretty good rule. Read the tiny ad, that's a pretty good rule, too.
It's like being on a deserted island - only with gardens and natural buildings in Montana