Dark-field imaging is a scattering-based X-ray imaging method that can be performed with laboratory X-ray tubes. The possibility to obtain information about unresolvable structures has already seen a lot of interest for both medical and material science applications. Unlike conventional X-ray attenuation, orientation dependent changes of the dark-field signal can be used to reveal microscopic structural orientation. To date, reconstruction of the three-dimensional dark-field signal requires dedicated, highly complex algorithms and specialized acquisition hardware. This severely hinders the possible application of orientation-dependent dark-field tomography. In this paper, we show that it is possible to perform this kind of dark-field tomography with common Talbot-Lau interferometer setups by reducing the reconstruction to several smaller independent problems. This allows for the reconstruction to be performed with commercially available software and our findings will therefore help pave the way for a straightforward implementation of orientation-dependent dark-field tomography.