We study the Landau damping of ferromagnetic magnons in Fe, Co, and Ni as the dimensionality of the system is reduced from three to two. We resort to the ab initio linear response time-dependent density functional theory in the adiabatic local spin density approximation. The numerical scheme is based on the Korringa-Kohn-Rostoker Green's function method. The key points of the theoretical approach and the implementation are discussed. We investigate the transition metals in three different forms: bulk phases, freestanding thin films, and thin films supported on a nonmagnetic substrate. We demonstrate that the dimensionality trends in Fe and Ni are opposite: in Fe the transition from bulk bcc crystal to Fe/Cu(100) film reduces the damping whereas in Ni/Cu(100) film the attenuation increases compared to bulk fcc Ni. In Co, the strength of the damping depends relatively weakly on the sample dimensionality. We explain the difference in the trends on the basis of the underlying electronic structure. The influence of the substrate on the spin-wave damping is analyzed by employing Landau maps representing wave-vector-resolved spectral density of the Stoner excitations.