I usually use the sponges from Trader Joes and a Dobie pad for cleaning dishes. They've been working great but the Dobie pads obviously aren't compostable (I think the Trader Joe sponges are - we just started trying to compost them). I really don't like throwing out the sponges every week but never felt comfortable with putting them in the dishwasher to clean since I worried about them breaking down and releasing toxins.
I ordered and just received the 3M Scotch-Bright Greener Clean Sponges and Scour Pad. The sponge states that it's compostable and the scour pad made from fibers of the agave plant and I assume it's compostable as well. Interested to try them. I hope I like them because that's one less thing heading towards a landfill if I do!
Scotch-Brite Greener Clean Absorbent Sponges