In this paper, we investigate the three-dimensional (3D) motion of a test particle in a stationary, axially symmetric spacetime around a central compact object, under the influence of a radiation field. To this aim, we extend the two-dimensional version of the Poynting-Robertson effect in general relativity that was developed in previous studies. The radiation flux is modeled by photons which travel along null geodesics in the 3D space of a Kerr background and are purely radial with respect to the zero angular momentum observer (ZAMO) frames. The 3D general relativistic equations of motion that we derive are consistent with the classical (i. e., non-general relativity) description of the Poynting-Robertson effect in three dimensions. The resulting dynamical system admits a critical hypersurface, on which radiation force balances gravity. Selected test particle orbits are calculated and displayed, and their properties are described. It is found that test particles approaching the critical hypersurface at a finite latitude and with nonzero angular moment are subject to a latitudinal drift and asymptotically reach a circular orbit on the equator of the critical hypersurface, where they remain at rest with respect to the ZAMO. On the contrary, test particles that have lost all their angular momentum by the time they reach the critical hypersurface do not experience this latitudinal drift and stay at rest with respect to the ZAMO at fixed nonzero latitude.