We demonstrate three-dimensional (3D) super-resolution in live multicellular organisms using structured illumination microscopy (SIM). Sparse multifocal illumination patterns generated by a digital micromirror device (DMD) allowed us to physically reject out-of-focus light, enabling 3D subdiffractive imaging in samples eightfold thicker than had been previously imaged with SIM. We imaged samples at one 2D image per second, at resolutions as low as 145 nm laterally and 400 nm axially. In addition to dual-labeled, whole fixed cells, we imaged GFP-labeled microtubules in live transgenic zebrafish embryos at depths >45 μm. We captured dynamic changes in the zebrafish lateral line primordium and observed interactions between myosin IIA and F-actin in cells encapsulated in collagen gels, obtaining two-color 4D super-resolution data sets spanning tens of time points and minutes without apparent phototoxicity. Our method uses commercially available parts and open-source software and is simpler than existing SIM implementations, allowing easy integration with wide-field microscopes.