[Unfortunately I didn’t take any photos… all I have left of the repair are fading memories, an email for the replacement part in January 2015, and a working dishwasher.]
I figured there must be some inlet valve that broke. Time to take a look.
I removed the one remaining loose screw holding the washer into the cabinet, and slid it forward. Clearly, all the good stuff is on the back and bottom sides, so I had to pull it completely out, and actually flip it over on its top.
I traced the route of water into the machine and found the inlet valve. I unplugged the spade plugs going to it, disconnected the water inlet and outlet connections, and took it down to the workbench.
I tried applying 120v AC to the terminals, and nothing happened — no clicking, nada. So, time to get a replacement.
Unfortunately, my usual go-to site for replacement parts, RepairClinic.com, did not have it. But, I found it on Sears Parts Direct.
After ordering the part and waiting a few days, it was easy enough to connect it in place of the old one, flip the dishwasher back over, and slide it into the cabinet. I took this as an opportunity to adjust the leg height and install two new mounting screws, so it’s no longer a bit wobbly. A quick test showed that the unit worked again. Hurray for our team!