Thanks to everyone who helped me out with this. I now have my Chess 960 position generator working on a ATTiny85 and a SCDQ5584 LED display. There are some libraries for driving the displays, but using the Tiny85 required me to save space and "iterate over bits on one pin." I may have re-written shiftOut() in the process, but it works and I understand how it's working, so that is nice. It's really cool to get this project down to the Tiny from the Atmega328 I originally wrote it on.
So thanks very much!
(Edit: and Karma points given out, too)
Jimmy