The effect of annealing on 1 mum thick single and multilayer amorphous carbon (a-C) films prepared by filtered cathodic arc is investigated. Single layer films, with a sp(2) to sp(3) bonding fraction of approximately 50% increase their level of compressive stress following annealing. Multilayer films-consisting of alternating layers of high sp(3) fraction (tetrahedral amorphous carbon, ta-C) and intermediate sp(3) fraction show a decrease in compressive stress following annealing. Using cross-sectional transmission electron microscopy, we show that the single layer films and the intermediate sp(3) layers in the multilayer films develop a strong preferred orientation with graphite-like layers aligned perpendicular to the film surface. The ta-C layers in the multilayer films develop the opposite preferred orientation near their top interfaces. We conclude that these preferred orientation effects are linked to the stress profile of the structures. We propose an underlying mechanism for the annealing effects of a-C films based on ab initio calculations. In order to minimize total energy, intermediate sp(3) films will either decrease their sp(3) fraction and generate stress or increase their sp(3) fraction and relieve stress. On the other hand, high sp(3) films retain their high sp(3) fraction following annealing.