We have constructed a microscope that provides three-dimensional (3D) super-resolution in live multicellular organisms using structured illumination microscopy (SIM). We use sparse multifocal illumination patterns generated by a digital micromirror device (DMD) to physically reject out-of-focus light, enabling 3D subdiffractive imaging in samples eightfold thicker than previously possible. The microscope can image samples at a rate of one 2D image per second, at resolutions as low as 145 nm laterally and 400 nm axially. We demonstrate the applicability of the technique to a wide variety of samples, including: dual-labeled whole fixed cells, GFP-labeled microtubules in live transgenic zebrafish embryos, and GFP-histones in nematode embryos. We also capture dynamic changes in the zebrafish lateral line primordium and observing 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 (we estimate the microscope can be constructed for < $100k, significantly cheaper than most confocal microscopes). The software is available at code.google.com/p/msim/.