AbstractGrating-based X-ray interferometry offers vast potential for imaging materials and tissues that are not easily visualised using conventional X-ray imaging. Tomographic reconstruction based on X-ray interferometric data provides not only access to the attenuation coefficient of an object, but also the refractive index and information about ultra-small-angle scattering. This improved functionality comes at the cost of longer measurement times because existing projection-based signal extraction algorithms require not only a single measurement per projection angle but several with precise grating movements in between. This obstacle hinders the adaptation of grating-based interferometry into a continuously rotating gantry. Several solutions to this problem have been proposed but all suffer from major drawbacks. We present results using an iterative reconstruction algorithm working directly on the interferograms. The suggested direct approach enables improved image quality, since interpolations and unnecessary assumptions about the object are circumvented. Our results demonstrate that it is possible to successfully reconstruct the linear attenuation coefficient, the refractive index and the linear diffusion coefficient, which is a measure related to ultra-small-angle scattering, using a single measurement per projection angle and without any grating movements. This is a milestone for future clinical implementation of grating-based phase-contrast and dark-field contrast X-ray computed tomography.