Super-resolution structured illumination microscopy (SR-SIM) can be conducted at video-rate acquisition speeds when combined with high-speed spatial light modulators and sCMOS cameras, rendering it particularly suitable for live-cell imaging. If, however, three-dimensional (3D) information is desired, the sequential acquisition of vertical image stacks employed by current setups significantly slows down the acquisition process. In this work, we present a multiplane approach to SR-SIM that overcomes this slowdown via the simultaneous acquisition of multiple object planes, employing a recently introduced multiplane image splitting prism combined with highspeed SIM illumination. This strategy requires only the introduction of a single optical element and the addition of a second camera to acquire a laterally highly resolved 3D image stack. We demonstrate the performance of multiplane SIM by applying this instrument to imaging the dynamics of mitochondria in living COS-7 cells.