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Understanding phonons in α-RuCl_{3} is critical to analyze the controversy around the observation of the half-integer thermal quantum Hall effect. While many studies have focused on the magnetic excitations in α-RuCl_{3}, its vibrational excitation spectrum has remained relatively unexplored. We investigate the phonon structure of α-RuCl_{3} via inelastic neutron-scattering experiments and density-functional-theory calculations. Our results show excellent agreement between experiment and first-principles calculations. After validating our theoretical model, we extrapolate the low-energy phonon properties. We find that the phonons in α-RuCl_{3} that either propagate or vibrate in the out-of-plane direction have significantly reduced velocities and therefore have the potential to dominate the observability of the elusive half-integer plateaus in the thermal Hall conductance. In addition, we use low-energy interlayer phonons to resolve the low-temperature stacking structure of our large crystal of α-RuCl_{3}, which we find to be consistent with that of the R3[over ¯] space group, in agreement with neutron diffraction.