Organic-inorganic lead-halide perovskites have received a revival of interest in the past few years as a promising class of materials for photovoltaic applications. Despite recent extensive research, the role of cations in defining the high photovoltaic performance of these materials is not fully understood. Here, we conduct nonadiabatic molecular dynamics simulations to study and compare nonradiative hot carrier relaxation in three lead-halide perovskite materials: CH3NH3PbI3, HC(NH2)2PbI3, and CsPbI3. It is found that the relaxation of hot carriers to the band edges occurs on the ultrafast time scale and displays a strong quantitative dependence on the nature of the cations. The obtained results are explained in terms of electron-phonon couplings, which are strongly affected by the atomic displacements in the Pb-I framework triggered by the cation dynamics.