Oldstoveguy was right on target.
After soaking the part with the filters, I took it outside, stuck a high pressure nozzle on the garden hose, and sprayed the filters. I finally got the gunk out.
I just reassembled everything (left out one part -- either a "Plate - air inlet cover" or "Gasket - cover plate") and our 22-year old dishwasher's working again.
Thanks to all for the help. And where can I send a donation to support this forum?