We describe a computational light-sheet microscope designed for hyperspectral acquisition at high spectral resolution. The fluorescence light emitted from the full field-of-view is focused along the entrance slit of an imaging spectrometer using a cylindrical lens. To acquire the spatial dimension orthogonal to the slit of the spectrometer, we propose to illuminate the specimen with a sequence of structured light patterns and to solve the image reconstruction problem. Beam shaping is obtained simply using a digital micromirror device in conjunction with a traditional selective plane illumination microscopy setup. We demonstrate the feasibility of this method and report the first results in vivo in hydra specimens labeled using two fluorophores.