Hyperspectral imaging (HSI) records a series of two-dimensional (2D) images for different wavelengths to provide the chemical fingerprint at each pixel. Combining HSI with a tomographic data acquisition method, we can obtain the chemical fingerprint of a sample at each point in three-dimensional (3D) space. The so-called 3D HSI typically suffers from low imaging throughput due to the requirement of scanning the wavelength and rotating the beam or sample. In this paper we present an optical system which captures the entire four-dimensional (4D), i.e., 3D structure and 1D spectrum, dataset of a sample with the same throughput of conventional HSI systems. Our system works by combining snapshot projection optical tomography (SPOT) which collects multiple projection images with a single snapshot, and Fourier-transform spectroscopy (FTS) which results in superior spectral resolution by collecting and processing a series of interferogram images. Using this hyperspectral SPOT system we imaged the volumetric absorbance of dyed polystyrene microbeads, oxygenated red blood cells (RBCs), and deoxygenated RBCs. The 4D optical system demonstrated in this paper provides a tool for high-throughput chemical imaging of complex microscopic specimens.