Results are presented from a computational study of the flow over a forward-facing step in a plane channel. The aim of the study is to gain better insight into the three-dimensionality that is typically observed in the separation region of flows over steps and ribs, and in similar configurations. We consider laminar flow at a Reynolds number of 330, based on step height and bulk velocity of the oncoming flow, and the step is assumed to be infinitely extended in the spanwise direction. High-resolution simulations are undertaken using a mixed spectral/spectral-element code. Moreover, a linear stability study of the flow at the step is performed. The results show that, in the case considered, the three-dimensionality is not related to some absolute instability of the separation bubble in front of the step; rather, it is a sensitive reaction of the flow to three-dimensional perturbations present in the oncoming stream. It is demonstrated that disturbance amplitudes of less than 1% of the mean flow (at, say, 10 step heights ahead of the step) already suffice to produce a visibly three-dimensional structure of the separation zone. If the disturbance level is systematically decreased, the three-dimensional state evolves to an almost two-dimensional recirculation. Here, the key finding is that the intensity of the flow response is proportionate to the amplitude of the inflow disturbance, meaning that the breakup of the flow in the step region is a linear (i.e. small) perturbation of the two-dimensional base flow. A comparison of the present simulation results with experimental data shows close agreement concerning, for example, the flow topology in the step region, and the spanwise spacing of the characteristic streaks that form further downstream.