  We study the box dimensions of self-affine sets in $\mathbb{R}^3$ which are
generated by a finite collection of generalised permutation matrices. We obtain
bounds for the dimensions which hold with very minimal assumptions and give
rise to sharp results in many cases. There are many issues in extending the
well-established planar theory to $\mathbb{R}^3$ including that the principal
planar projections are (affine distortions of) self-affine sets with overlaps
(rather than self-similar sets) and that the natural modified singular value
function fails to be sub-multiplicative in general. We introduce several new
techniques to deal with these issues and hopefully provide some insight into
the challenges in extending the theory further.
