  Axion dark matter can resonantly convert into photons in the magnetospheres
of neutron stars (NSs). It has recently been shown that radio observations of
nearby NSs can therefore provide a highly sensitive probe of the axion
parameter space. Here we extend existing calculations by performing the first
three-dimensional computation of the photon flux, taking into account the
isotropic phase-space distribution of axions and the structure of the NS
magnetosphere. In particular, we study the overall magnitude of the flux and
its possible time variation. We find that overall signal strength is robust to
our more realistic analysis. In addition, we find that the variance of the
signal with respect to the NS rotation is washed out by the additional
trajectories in our treatment. Nevertheless, we show that SKA observations
towards J0806.4-4123 are sensitive to $g_{a\gamma\gamma}\sim
3\times10^{-13}\mathrm{\,GeV^{-1}}$ at $m_a\sim 7\times10^{-6}\mathrm{\,eV}$,
even when accounting for Doppler broadening. Finally, we provide the necessary
code to calculate the photon flux for any given NS system.
