X-ray computed tomography (CT) can distinguish different materials according to their absorption characteristics. The hyperspectral X-ray CT (HXCT) system proposed in the present work reconstructs each voxel according to its X-ray absorption spectral characteristics. In contrast to a dual-energy or multi-energy CT system, HXCT employs cadmium telluride (CdTe) as the x-ray detector, which provides higher spectral resolution and separate spectral lines according to the material's photon-counter working principle. In this paper, a specimen containing ten different polymer materials randomly arranged was adopted for material identification by HXCT. The filtered back-projection algorithm was applied for image and spectral reconstruction. The first step was to sort the individual material components of the specimen according to their cross-sectional image intensity. The second step was to classify materials with similar intensities according to their reconstructed spectral characteristics. The results demonstrated the feasibility of the proposed material identification process and indicated that the proposed HXCT system has good prospects for a wide range of biomedical and industrial nondestructive testing applications.