KCuCl₃ is an S = 1/2 magnetic insulator with a singlet ground state and a finite spin excitation gap. Above the gap, dispersive triplet excitation modes propagate in the whole reciprocal space. From single-crystal inelastic neutron investigations the three-dimensional coupling scheme is rationalized in the framework of a dimer Heisenberg model, and related to the structural features of KCuCl₃. The experimental and theoretical characterization presented completes earlier works on the compound under investigation, providing also higher-order expressions for the singlet-triplet dispersion relation. The latter may also be of relevance for the parent quantum systems TlCuCl₃ and NH₄CuCl₃, albeit at different coupling ratios with respect to KCuCl₃.