I have to admit that I took some inspiration for this from Greg Zumwalt's wonderful 3D Printed Christmas Billboard. So in keeping with the idea of having a simple mechanical mechanism for each dot in a flip dot display I designed this tri-state display element.
Like the bi-state flip square shown previously, this design uses magnets to "lock" the display into a stable position when a dot is facing forwards. (Note that the prism with the dots is shown on the right photo above upside down so that the magnets can be seen.) In addition when in the locked position, the backward facing vertex of the prism provides a perfect lever for an actuator to push the dot (in either direction) to the next state. Here's how it works.
It's even simpler mechanically than the push-push flip square. There is still the issue of determining the current state of the visible dot. I think that this could be easily solved with some strategically embedded magnets and a couple of hall-effect sensors.