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 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.