In this Letter, we present a method for hyperspectral imaging of three-dimensional objects. A compressive sensing approach is utilized to remedy the acquisition effort required to capture the large amount of data. The spectral dimension is compressively sensed by means of a liquid crystal-based encoder, and the volumetric data are captured using a synthetic aperture integral imaging setup. We demonstrate reconstruction of spectro-volumetric tesseracts with hundreds of spectral bands at different depths without compromise of spatial resolution.