We present a new method to model spin-wave excitations in magnetic solids, based on the Liouville-Lanczos approach to time-dependent density functional perturbation theory. This method avoids computationally expensive sums over empty states and naturally deals with the coupling between spin and charge fluctuations, without ever explicitly computing charge-density susceptibilities. Spin-wave excitations are obtained with one Lanczos chain per magnon wave-number and polarization, avoiding the solution of the linear-response problem for every individual value of frequency, as other state-of-the-art approaches do. Our method is validated by computing magnon dispersions in bulk Fe and Ni, resulting in agreement with previous theoretical studies in both cases, and with experiment in the case of Fe. The disagreement in the case of Ni is also comparable with that of previous computations.