Small-angle x-ray scattering tensor tomography provides three-dimensional information on the unresolved material anisotropic microarchitecture, which can be hundreds of times smaller than an image pixel. We develop a direct filtered back-projection method based on algebraic filters that enables rapid tensor-tomographic reconstructions and is a few orders of magnitude faster compared to established techniques, given the same computational resources. We demonstrate the accuracy of the method on experimental data for a fiber-reinforced material sample. The achieved acceleration may pave the way toward the investigation of multiple large samples as well as rapid control and feedback during in situ tensor-tomographic experiments, opening perspectives for the understanding of the fundamental link between functional material properties and microarchitecture.