Why is it hard to extend the Feynman Checkerboard to more than 1+1 dimensions?

Dear mtrencseni, the very page you quoted has an answer in the very following sentences:

Two distinct classes of extensions exist, those working with a fixed underlying lattice (19,20) and those that embed the two dimensional case in higher dimension (21,22). The advantage of the former is that the sum-over-paths is closer to the non-relativistic case, however the simple picture of a single directionally independent speed of light is lost. In the latter extensions the fixed speed property is maintained at the expense of variable directions at each step.

There is clearly no reason why similar mathematical curiosities should exist in higher dimensions. Nevertheless people tried to find a higher-dimensional counter along two different competing lines. Both of them remain highly inconclusive, to put it very generously.

The most obvious reason why the Feynman checkerboard is hard in higher dimensions is that the Dirac equation has many components and the individual spinor components cannot associated with any simple discrete properties of a lattice, such as the direction where the particle moves in the first step. The fact that the geometry of direction on a line is so simple is the reason why the Feynman checkerboard in 1+1 dimensions, and only in 1+1 dimensions, may be used as a bookkeeping device for the helicity components etc.

Also, the higher-dimensional lattices don't include lines that move "exactly by the speed of light" (45 degrees) in generic directions. That's also different from 1+1 dimensions where the Southwest and Southeast directions are the only two light-like directions. This simple fact makes it hard to obtain a relativistic theory from a particle moving on a lattice.

There's an interesting paper on the subject, written at a lower level than some, on my website here.

It notes that there are issues with the particle in 3 dimensions apparently being superluminal with a speed of at least $\sqrt{3}\;c$.

In 1+1 dimensional spacetime you can split the operator $d^{2}/dt^{2} - d^{2}/dx^{2}$ in a very simple way into a product: $(d/dt - d/dx)(d/dt + d/dx)$. This is not possible in higher dimensions.