While a gyro in theory gives you information to get the answer, in practise the angle will have so much error after a few seconds as to be useless. MEMs gyros put out a signal indicating rotation rate. This needs to be integrated to get angle. However, offset, drift, and other errors will continuously accumulate in the integral. For most MEMs gyros, the result is useless after a second or a few seconds.
Note that real mechanical gyros don't have this problem since they inherently give a angular position signal, not angular rate. However, they are big, klunky, take significant power, are expensive, and need to be "caged" for a minute to 10s of minutes with no motion to calibrate them before use.
If the rotation rate is reasonably constant, a tick once per revolution is good enough. I've worked on several rotating LED signs where that was the only information on position. The system assumed each rotation would take the same time as the previous rotation. The display did get stretched during turn on when the motor was accellerating, but it worked quite well at steady state.
Just this spring I advised a high shool student that wanted to do something similar. I gave him a single opto-interrupter, which he used to compute the time per rotation and generate the pixel clock for the next rotation. His motor was a large fan lying on its side on the floor, with the board battery powered and mounted in the middle. It spelled out "2011 SENIOR PROJECT" using 7 LEDs, and everyone was quite suitably impressed.
If the shaft is horizontal you can use a accellerometer to measure rotation speed and get some idea of the current angle by watching gravity.
Depending on how the motor works, you could possibly have the motor driving electronics report position regularly. If it's a brushless DC and you have access to the controller, this information is available.
Communicating to the rotating board from outside can be done several ways. In the rotating LED sign, we used a hollow shaft with a IR LED on the fixed end and a photodiode on the rotating end. This was used as a one way data stream at about 1Mbit/s. Getting power to the rotating board is also interesting. The high school student used batteries. That was easy and worked fine for his presentation. One version of the LED sign needed 70W and had to run continuously, so that wasn't possible. We used a transformer where the primary and secondary could rotate with respect to each other. This was driven by a switcher at high frequency. A small secondary winding on the fixed side was used for feedback, and the rotating secondary was regulated well enough. There was a additional switching power supply on the rotating board to trim the power loss in driving the RGB LEDs to as little as possible.