A full orbit code is used to study the collisionless transport of energetic ions in spherical and conventional tokamak plasmas with rotating magnetic island chains. Such islands are associated with neoclassical tearing modes, which occur in the vicinity of mode rational surfaces in many tokamaks, and, in the case of the conventional tokamaks DIII-D and Asdex-Upgrade, have been observed to enhance fast particle transport. Magnetic island chains are represented as single-helicity perturbations to the equilibrium poloidal magnetic flux. Loss rates of neutral beam-injected fast ions are computed for a range of perturbation amplitudes and mode frequencies in plasma equilibria representative of those in the mega-ampère spherical tokamak and a large conventional tokamak. It is found that most of the island-induced losses are of ions that are born into passing orbits. For some injection scenarios the loss rate is shown to be a sensitive function of mode frequency, although the interaction of the mode with passing particles is non-resonant. The loss rate is generally lower for island chains rotating toroidally in the direction of beam injection than it is for island chains that are either static or rotating counter to the beam direction.