We report on significant advances and new results concerning a recently developed method for grating-based hard x-ray phase tomography. We demonstrate how the soft tissue sensitivity of the technique is increased and show in vitro tomographic images of a tumor bearing rat brain sample, without use of contrast agents. In particular, we observe that the brain tumor and the white and gray brain matter structure in a rat's cerebellum are clearly resolved. The results are potentially interesting from a clinical point of view, since a similar approach using three transmission gratings can be implemented with more readily available x-ray sources, such as standard x-ray tubes. Moreover, the results open the way to in vivo experiments in the near future.