  In the field of frustrated magnetism, Kitaev models provide a unique
framework to study the phenomena of spin fractionalization and emergent lattice
gauge theories in two and three spatial dimensions. Their ground states are
quantum spin liquids, which can typically be described in terms of a Majorana
band structure and an ordering of the underlying $\mathbb{Z}_2$ gauge
structure. Here we provide a comprehensive classification of the "gauge
physics" of a family of elementary three-dimensional Kitaev models, discussing
how their thermodynamics and ground state order depends on the underlying
lattice geometry. Using large-scale, sign-free quantum Monte Carlo simulations
we show that the ground-state gauge order can generally be understood in terms
of the length of elementary plaquettes -- a result which extends the
applicability of Lieb's theorem to lattice geometries beyond its original
scope. At finite temperatures, the proliferation of (gapped) vison excitations
destroys the gauge order at a critical temperature scale, which we show to
correlate with the size of vison gap for the family of three-dimensional Kitaev
models. We also discuss two notable exceptions where the lattice structure
gives rise to "gauge frustration" or intertwines the gauge ordering with
time-reversal symmetry breaking. In a more general context, the thermodynamic
gauge transitions in such 3D Kitaev models are one of the most natural settings
for phase transitions beyond the standard Landau-Ginzburg-Wilson paradigm.
