It was embarrassingly cheap and easy:
I bought a giant wool sweater at the thrift store (I specifically looked for something really thick, labeled handwash or dry-clean only, so it would felt) that had a crew/round neckline.
Then I cut off the sleeves, in line with the body of the sweater. I flipped it inside out and stitched the sleeves shut and the bottom shut. So now I had a bag with an opening at the top. I threaded a ribbon through the stitches around the neck to form a drawstring.
Threw the whole thing in the machine, felted it until it looked thick enough, rinsed it and air-dried it. (I plan to lanolinize it, but haven't gotten to it yet - it still works fine.)
I hang it on a doorknob by the ribbon, and the diapers go in the bag through the neck opening. On laundry day, I just loosen the drawstring and dump the diapers into the machine. But the one change I think I'm going to make is to put a zipper in the bottom seam, so on wash day I can just unzip it and they'll all tumble out at once, no shaking.
Size-wise, I think it's about 30 inches high by 18 inches wide. It's big enough to hold three days of diapers of my 5 month old.