Large amplitude quantum oscillations of magnetic anisotropy in Fe films grown on Ag surfaces of high step density are reported. Distortion of the film structure by the steps is found to be responsible for the anisotropy modification which oscillates with increasing film thickness. The larger the distorted fraction of the film volume, the larger the observed amplitude of the anisotropy oscillations. Such dependence of the anisotropy oscillations on the step density offers an easy tool to manipulate the strength of the anisotropy and orientation of the easy magnetization axis.