  Coupled-wire constructions have proven to be useful tools to characterize
Abelian and non-Abelian topological states of matter in two spatial dimensions.
In many cases, their success has been complemented by the vast arsenal of other
theoretical tools available to study such systems. In three dimensions,
however, much less is known about topological phases. Since the theoretical
arsenal in this case is smaller, it stands to reason that wire constructions,
which are based on one-dimensional physics, could play a useful role in
developing a greater microscopic understanding of three-dimensional topological
phases. In this paper, we provide a comprehensive strategy, based on the
geometric arrangement of commuting projectors in the toric code, to generate
and characterize coupled-wire realizations of strongly-interacting
three-dimensional topological phases. We show how this method can be used to
construct pointlike and linelike excitations, and to determine the topological
degeneracy. We also point out how, with minor modifications, the machinery
already developed in two dimensions can be naturally applied to study the
surface states of these systems, a fact that has implications for the study of
surface topological order. Finally, we show that the strategy developed for the
construction of three-dimensional topological phases generalizes readily to
arbitrary dimensions, vastly expanding the existing landscape of coupled-wire
theories. Throughout the paper, we discuss $\mathbb{Z}^{\,}_m$ topological
order in three and four dimensions as a concrete example of this approach, but
the approach itself is not limited to this type of topological order.
