High-temperature copper oxide superconductors consist of stacked CuO2 planes, with electronic band structures and magnetic excitations that are primarily two-dimensional1,2, but with superconducting coherence that is three-dimensional. This dichotomy highlights the importance of out-of-plane charge dynamics, which has been found to be incoherent in the normal state3,4 within the limited range of momenta accessible by optics. Here we use resonant inelastic X-ray scattering to explore the charge dynamics across all three dimensions of the Brillouin zone. Polarization analysis of recently discovered collective excitations (modes) in electron-doped copper oxides5-7 reveals their charge origin, that is, without mixing with magnetic components5-7. The excitations disperse along both the in-plane and out-of-plane directions, revealing its three-dimensional nature. The periodicity of the out-of-plane dispersion corresponds to the distance between neighbouring CuO2 planes rather than to the crystallographic c-axis lattice constant, suggesting that the interplane Coulomb interaction is responsible for the coherent out-of-plane charge dynamics. The observed properties are hallmarks of the long-sought 'acoustic plasmon', which is a branch of distinct charge collective modes predicted for layered systems8-12 and argued to play a substantial part in mediating high-temperature superconductivity10-12.