Imaging speed and spatial resolution are key factors in optical diffraction tomography (ODT), while they are mutually exclusive in 3D refractive index imaging. This paper presents a multi-harmonic structured illumination-based optical diffraction tomography (MHSI-ODT) to acquire 3D refractive index (RI) maps of transparent samples. MHSI-ODT utilizes a digital micromirror device (DMD) to generate structured illumination containing multiple harmonics. For each structured illumination orientation, four spherical spectral crowns are solved from five phase-shifted holograms, meaning that the acquisition of each spectral crown costs 1.25 raw images. Compared to conventional SI-ODT, which retrieves two spectral crowns from three phase-shifted raw images, MHSI-ODT enhances the imaging speed by 16.7% in 3D RI imaging. Meanwhile, MHSI-ODT exploits both the 1st-order and the 2nd-order harmonics; therefore, it has a better intensity utilization of structured illumination. We demonstrated the performance of MHSI-ODT by rendering the 3D RI distributions of 5 µm polystyrene (PS) microspheres and biological samples.