So the guy came out on Sunday and basically installed a new ice maker for us. Yay!
Today we discovered a leak in the water line supplying said new ice maker. Much stuff from the basement was tossed in sacrifice to the new ice maker. Fortunately, none of it was terribly important.
So I went off to get a replacement water line. The repair guy from Sunday had suggested replacing the plastic with copper at some point (something they didn't cover), so I got a copper line at the local Home Depot. A couple of hours later, with some grunting and sweating and crawling behind the fridge in there, I got the line installed. Had to go back and fix a leak at the supply valve but otherwise things look OK for now. I'll check for leaks again when I get home, just to be sure.