Coherent scattering of x-ray photons leads to the phenomenon of x-ray diffraction, which is widely used for determining atomic structure in materials science. A technique [x-ray diffraction computed tomography (CT)] is described, analogous to conventional CT, in which the x-ray diffraction properties of a stack of two-dimensional object sections may be imaged. The technique has been investigated using a first generation (single pencil beam) CT scanner to measure small angle coherent scatter, in addition to the customary transmitted radiation. Diffraction data from a standard CT performance phantom obtained with this new technique and with an x-ray diffractometer are compared. The agreement is satisfactory bearing in mind the poor momentum resolution of our apparatus. The dose and sensitivity of x-ray diffraction CT are compared with those of conventional transmission CT. Diffraction patterns of some biological tissues and plastics presented in a companion paper indicate the potential of x-ray diffraction CT for tissue discrimination and material characterization. Finally, possibilities for refinement of the technique by improving the momentum resolution are discussed.