There is growing interest in voxelated matter that is designed and fabricated voxel by voxel1–4. Currently, inkjet-based three-dimensional (3D) printing is the only widely adopted method that is capable of creating 3D voxelated materials with high precision1–4, but the physics of droplet formation requires the use of low-viscosity inks to ensure successful printing5. By contrast, direct ink writing, an extrusion-based 3D printing method, is capable of patterning a much broader range of materials6–13. However, it is difficult to generate multimaterial voxelated matter by extruding monolithic cylindrical filaments in a layer-by-layer manner. Here we report the design and fabrication of voxelated soft matter using multimaterial multinozzle 3D (MM3D) printing, in which the composition, function and structure of the materials are programmed at the voxel scale. Our MM3D printheads exploit the diode-like behaviour that arises when multiple viscoelastic materials converge at a junction to enable seamless, high-frequency switching between up to eight different materials to create voxels with a volume approaching that of the nozzle diameter cubed. As exemplars, we fabricate a Miura origami pattern14 and a millipede-like soft robot that locomotes by co-printing multiple epoxy and silicone elastomer inks of stiffness varying by several orders of magnitude. Our method substantially broadens the palette of voxelated materials that can be designed and manufactured in complex motifs.