We present a first principles implementation of the dynamic transverse magnetic susceptibility in the framework of linear response time-dependent density functional theory. The dynamic susceptibility allows one to obtain the magnon dispersion as well as magnon lifetimes for a particular material, which strongly facilitates the interpretation of inelastic neutron scattering experiments as well as other spectroscopic techniques. We apply the method to Fe, Ni, and Co and perform a thorough convergence analysis with respect to the basis set size, k-point sampling, spectral smearing, and unoccupied bands. In particular, it is shown that while the gap error (acoustic magnon energy at q = 0) is highly challenging to converge, the spin-wave stiffness and the dispersion relation itself are much less sensitive to convergence parameters. Our final results agree well with experimentally extracted magnon dispersion relations except for Ni, where it is well known that the exchange splitting energy is poorly represented in the local density approximation. We also find good agreement with previous first principles calculations and explain how differences in the calculated dispersion relations can arise from subtle differences in computational approaches.