The signals on my 1981 KZ 750 LTD are acting up. They either
Flash very slowly,
Flash too rapidly, or
Don't flash at all
The bulbs are fine so I'm thinking there must be a mechanism that controls the flashing (a "flash box"?)
Any thoughts?

 

Yep, odds are there is a turn signal relay either under a side cover or the seat. See if you can find a manual or ask a Kawasaki service department - or of course someone here - where it is. Then go to Autozone and buy a replacment. I put different (Buell Ulysses rear) turn signals on both my Kawasakis. They use a higher power bulb and would not flash, a $6.99 Autozone turn signal relay (flasher) took care of it just fine. Your old relay may be mechanical, the one to get from Autozone or where ever is an electronic (no load) relay like used when towing trailers (six bulbs instead of four).