Kitaev's honeycomb-lattice spin-1/2 model has become a paradigmatic example for Z2 quantum spin liquids, both gapped and gapless. Here we study the fate of these spin-liquid phases in differently stacked bilayer versions of the Kitaev model. Increasing the ratio between the interlayer Heisenberg coupling J and the intralayer Kitaev couplings Kx,y,z destroys the topological spin liquid in favor of a paramagnetic dimer phase. We study phase diagrams as a function of J/K and Kitaev coupling anisotropies using Majorana-fermion mean-field theory, and we employ different expansion techniques in the limits of small and large J/K. For strongly anisotropic Kitaev couplings, we derive effective models for the different layer stackings that we use to discuss the quantum phase transition out of the Kitaev phase. We find that the phase diagrams depend sensitively on the nature of the stacking and anisotropy strength. While in some stackings and at strong anisotropies we find a single transition between the Kitaev and dimer phases, other stackings are more involved. Most importantly, we prove the existence of two novel macrospin phases, which can be understood in terms of Ising chains that can be either coupled ferromagnetically or remain degenerate, thus realizing a classical spin liquid. In addition, our results suggest the existence of a flux phase with spontaneous interlayer coherence. We discuss prospects for experimental realizations.