I'm curious if this exists, but simply don't know. I'm thinking of euclidian, flat, 3d-space. Preferably filling with one type of solid. Next neighbours means shares a plane with. Here are my thoughts so far:
Tetraeders would be 4 sided solids, but cannot tile space (fill it completely). Reading through this, it appears that no regular, 4-sided, tiling polyhedra exist, though non regular ones are not mentioned.
I think if you take a diamond packing (every node is connected to it's four nearest neighbour & the whole affair is highly regular), you could take the middle point of each edge and contruct a plane so that the edge cuts the plane at a right angle, these planes should make up such a tessalation ... exceot that if i try to visualize the this (in my head, my drawing/modelling skills are not up to the task), I arrive at tetraeders again which we know won't work - but maybe my visualization is wrong.
I think that a deformed tetraeder (not all four sides equal) could do it, but I lack the skills to really prove or test this.
It turns out I should have read a few of the tabs I kept open before posting:
The Tetragonal Disphenoid Honeycomb is what I'm looking for. It appears my hunch about constructing it from a diamond grid was not so bad, if you don't start with a diamond grid but a bitruncated cubic honeycomb.