From what I remember from someone who used to post here (haven't seen her in a while), they put out pallets with dirt and grass in them for the dogs to do their "business".
You do have to tell the cruise line that you're bringing a service dog. They may limit the number on each cruise. (Obviously, on booboo's cruise they didn't!)
Check out "special services" or something similar on the cruise line's websites.
I agree with gerif, if only the children were as well behaved as the dogs!