As I'd been suspecting, the 5-pin plastic connector I'd been using to connect the periscope controller board to the main periscope logic board wasn't working out, the wires were too loose in there. So I pulled the wires out.
I picked up some "Continuous Jumpers" from my local Fry's Electronics. These have connections on the ends that slip nicely over the connector pins on the two boards. Unfortunately, they are daisy-chained together, so I have to cut them and solder them to the existing wires, but it's not a big deal.
This works better. I have this kind of connection on each end now, and it makes it easier to thread the wires through the wiring hole in the periscope.