X-ray diffraction can be used as the signal for tomographic reconstruction and provides a cross-sectional map of the crystallographic phases and related quantities. Diffraction tomography has been developed over the last decade using monochromatic x-radiation and an area detector. This paper reports tomographic reconstruction with polychromatic radiation and an energy sensitive detector array. The energy dispersive diffraction (EDD) geometry, the instrumentation and the reconstruction process are described and related to the expected resolution. Results of EDD tomography are presented for two samples containing hydroxyapatite (hAp). The first is a 3D-printed sample with an elliptical crosssection and contains synthetic hAp. The second is a human second metacarpal bone from the Roman-era cemetery at Ancaster, UK and contains bio-hAp which may have been altered by diagenesis. Reconstructions with different diffraction peaks are compared. Prospects for future EDD tomography are also discussed.