The traditional Fourier single-pixel imaging (FSI) utilizes cosinusoidal (grayscale) illumination patterns to reconstruct the image of an object. In this paper, we propose a universal mechanism in which FSI can be implemented via arbitrary periodic structured illumination not merely cosinusoidal illumination patterns. The proposed theory is verified by numerical simulations and optical experiments. In the experiment, we demonstrate the result of FSI with phase-shifting sawtooth (trapezoid, square) structured illumination modulated by a light projector. Moreover, in order to underline the high efficiency of the proposed mechanism, we perform FSI utilizing binary (square) illumination patterns generated directly by the digital micromirror devices (DMDs). Therefore, this mechanism can promote the high-performance and extensive applications of FSI in various scenarios.