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 toward J0806.4-4123 are sensitive to g(a gamma gamma) similar to 3 x 10(-13) GeV (-1) at m(a) 7 x 10(-6) eV, even when accounting for Doppler broadening. Finally, we provide the necessary code to calculate the photon flux for any given NS system https://github.com/mikaelLEROY/AxionNS_RayTracing.