The dishwasher won't drain. What can I do? There appears to be no obvious drain where I could use a snake to dislodge a clog.

I don't know if this will help, but it'll at least explain your problem a bit.

Your dishwasher doesn't drain like a sink or tub -- using gravity to let water go down a drain. Rather, it uses a pump to push water up a hose, generally connected to a sink or garbage disposal next to the dishwasher.

When dishwashers stop draining, it's usually not a clog, it's usually that the pump has stopped working.

When the dishwasher is trying to drain (in the middle of the wash cycle) does it make any unusual sounds?

Most likely it is your drain line located at the bottom of your pump which is causing the dishwasher not to drain.
Clear as much of the water out of the dishwasher as possible, use a shop vac if you have one.
Remove bottom cover from dishwasher and you will see the drain line at the bottom of the pump,remove the line, you will see a hard pipe were the line attaches remove this hard pipe with a set of pliers. Inside the hard line is a diaphragm , this is usually were rice and corn get lodged in the pipe. Remove debris and re-attach lines in reverse order.
Note: the hard line will have o-ring , inspect o-rings and replace if needed , this will ensure that there will be no leaks.
Hopefully this info helps you.