I got a Rob and Gyromite.
I used a high speed camera (480 fps or 8x the TV rate) to look at the actual output from the TV. I got the same codes as Tursi. The instructions are given one per frame, at 60 fps. There is no black frame after outputing the instruction. The test sequence that flashes the LED on the head is: 00000 000101 1101011 11111.
I would think that having the correct timing is pretty important (although I would guess that R.O.B. still works with 50Hz TV systems from other countries).
I think the problem with a digital TV, even if it has a 60 Hz refresh rate, is that it may think that consecutive analog frames are interlaced and tries to de-interlace them, what mixes 0 and 1 frames.
By the way, I also looked at Duck Hunt: when you press the trigger you get a black frame, next one that has only a white square on the duck, next back to the normal display. If you have two ducks: black frame, frame with white square for the first duck, frame with white square for the second duck, again frame with white square for the second duck, resume normal display. Interestingly, if I route the TV signal through a VCR, the increased delay is enough to stop the gun from detecting the ducks.