Here's a neighborhood scheme:
The notion is, I can just pick any ol' unitary 4x4 matrix, and apply it to a group of four states (every black dot represents one complex number in a big state-vector spanning the whole 2-d spacelike array). On even time steps I apply it in parallel to each group outlined in red, and on odd timesteps to blue. This way hopefully things "circulate" enough.
Here's a quick little js implementation. Click to generate a new random matrix. I made sure the random number generator is deterministic, so you'll see the same sequence of random matrices each time you reload --- the seed number should appear at the top left. I think #2 and #5, #6 are quite pretty.