Linear structured illumination microscopy (SIM) is a super-resolution microscopy technique that does not impose photophysics requirements on fluorescent samples. Multicolor SIM implementations typically rely on liquid crystal on silicon (LCoS) spatial light modulators (SLM’s) for patterning the excitation light, but digital micromirror devices (DMD’s) are a promising alternative, owing to their lower cost and higher speed. However, existing coherent DMD SIM implementations use only a single wavelength of light, limited by the lack of efficient approaches for solving the blazed grating effect for polychromatic light. We develop the requisite quantitative tools, including a closed form solution of the blaze and diffraction conditions, forward models of DMD diffraction and pattern projection, and a model of DMD aberrations. Based on these advances, we constructed a three-color DMD microscope, quantified the effect of aberrations from the DMD, developed a high-resolution optical transfer function measurement technique, and demonstrated SIM on fixed and live cells. This opens the door to applying DMD’s in polychromatic applications previously restricted to LCoS SLM’s.