A periodic $(n,k)$ coloring can be modified to give a non-periodic $(2n,2k)$ coloring. This is a modification of Anthony's construction without need of probabilities. Let the $n$ colors be $\{{1,2,\cdots ,n\}}$ and allow $n$ more colors $\{{-1,-2,,\cdots,-n\}}$ (This is just for ease of description). Start with the $(n,k)$ coloring and  draw imaginary circles (or spheres etc.) centered at the origin of radii $3.5,3.5^2,3.5^3,...$ This splits the lattice points into concentric zones. Alternate making the colors positive and negative according to the zone. The lattice points on a lattice line occur with equal spacing so eventually (as you travel along a line, increasing the distance from the origin) you will experience a full period (maybe several) all in a single positive zone, then several in a single negative zone. 

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I am not convinced that there are periodic ones, but I am not convinced there are not. The first construction with diagonal matrices (if I understand correctly) has some lines of slope $-1$  with just $1,2.$ A line of slope $-1$ going through a point labelled $0$ would seem to have all points labelled $0.$ I did not look at the other constructions, and may misunderstand the first one.