In this paper, we examine the effect that electromagnetic dispersion has on the motion of an electron in a relativistically strong plane wave. We obtain an analytic solution for the electron momentum and check this solution against direct numerical integration of the equations of motion. The solution shows that even a relatively small difference between the phase velocity of the wave, vp, and the speed of light, c, can significantly alter the electron dynamics if the normalized wave amplitude a0 exceeds √2c/(vp-c). At this amplitude, the maximum longitudinal electron momentum scales only linearly with a0, as opposed to a02. We also show that at this amplitude the impact of an accelerating longitudinal electric field and electron pre-acceleration is negated by the superluminous phase velocity of the wave. This has implications for the potential of Direct Laser Acceleration of electrons. We point out that electromagnetic dispersion can arise from both propagation in a plasma and from propagating the laser in what is effectively a wave-guiding structure, and that this latter source of dispersion is likely to be more significant.