Today I put all the pins in their sockets and verified that the individual driver circuits worked.
I also had a problem with a look-up table jumping off the memory page (resulting in a jump somewhere in the middle of the ISR, giving totally unpredictable behaviour), so I changed the firmware somewhat. Instead of using a lookup table, I 'seed' the initial color values and then just add 3 to that number at each step. No lookup table needed, and I get pretty much every color available.
Now I have hooked it all up and inserted the RGB LEDs sideways into the pieces of plexiglass. It looks stunningly good. I'm even more excited about this project now!