  In the search for experimental realizations of bond-anisotropic Kitaev
interactions and resulting spin-liquid phases, the layered magnet
$\alpha$-RuCl$_3$ is a prime candidate. Its modelling typically involves
Heisenberg, Kitaev, and symmetric off-diagonal $\Gamma$ interactions on the
two-dimensional honeycomb lattice. However, recent neutron-scattering
experiments point towards a sizeable magnetic interlayer coupling. Here we
study three-dimensional exchange models for $\alpha$-RuCl$_3$, for both
possible $R\bar{3}$ and $C2/m$ crystal structures. We discuss the symmetry
constraints on the interlayer couplings, construct minimal models, and use them
to compute the magnetic mode dispersion and the dynamical spin structure
factor, in both the zero-field zigzag phase and the paramagnetic high-field
phase. Our predictions for the interlayer mode dispersion shall guide future
experiments; they also call for a reevaluation of the quantitative model
parameters relevant for $\alpha$-RuCl$_3$.
