Zombie spawners do not seem to work for me. I have tried setting the world to peaceful and back. Exiting and reentering. Increasing the radius of the dungeon to 9x9. Ensuring it is pitch dark. Ensuring the spinning Mob is indeed a Zombie. Nothing seems to work.

Another detail to point out is that this only seems to happen on Beta worlds. So any world that I have created that has the experimental gameplay activated Zombie spawners do not work for me. I do not know about any other mob spawners as I have only ever found naturally spawned Zombie spawners.

Any worlds I create that are not using the experimental gameplay seem to be fine and Zombie spawners are working fine on them. 

I am not sure if this is the correct place to post about potential glitches found in the beta of the game so apologies if I am in the wrong place. Hope this gets resolved in the next patch.