We discuss theoretical, experimental and numerical aspects of several new techniques for quantitative phase-contrast tomography using, for example, unfiltered radiation from a polychromatic X-ray microfocus source. The proposed algorithms allow one to reconstruct the three-dimensional distribution of complex refractive index in a sample consisting of one or more constituent materials, given one or more projection images per view angle. If the sample is weakly absorbing or consists predominantly of a single material, these reconstruction algorithms can be simplified and fewer projections may be required for an unambiguous quantitative reconstruction of the spatial distribution of the refractive index. In the case of weakly absorbing samples, the reconstruction algorithm is shown to be achromatic and stable with respect to high-spatial-frequency noise, in contrast to conventional tomography. A variation of the algorithm exploits the natural combination of binary tomography with a phase-retrieval method that makes explicit use of the single-material nature of the sample. Such consistent use of a priori knowledge dramatically reduces the number of required projections, implying significantly reduced dose and scanning time when compared to most alternative phase-contrast tomography methods. Experimental demonstrations are also given, using data from a point-projection X-ray microscope. The refractive index distribution, in test samples of both a polymer fibre scaffold and an adult mouse, is accurately reconstructed from polychromatic phase-contrast data. Applications of the new techniques to rapid non-destructive testing in materials science and biomedical imaging are considered.