An efficient scheme is presented to compute the transverse magnetic susceptibility within time-dependent density functional theory from which magnon dispersions can be extracted. The scheme makes use of maximally localized Wannier functions in order to interpolate the band structure onto a fine k mesh in order to converge sums on the first Brillouin zone. The gap error in the magnon dispersion at Γ, numerically violating Goldstone's theorem, is analyzed and a correction scheme is devised that can be generalized to systems where Goldstone's theorem does not apply. The method is applied to the computation of the magnon dispersion of bulk bcc iron and fcc nickel.