X-ray phase-contrast imaging can display subtle differences in low-density materials (e.g. soft tissues) more readily than conventional x-ray imaging. However, producing x-ray phase images requires significant spatial coherence of the beam which in turn requires highly specialized sources such as synchrotrons, small and low power microfocus sources, or complex procedures, such as multiple exposures with several carefully stepped precision gratings. To find appropriate approaches for producing x-ray phase-contrast imaging in a clinically meaningful way, we employed a grating-free method that utilized a low-cost, coarse wire mesh and simple processing. This method relaxes the spatial coherence constraint and allows quantitative phase retrieval for not only monochromatic but also polychromatic beams. We also combined the mesh-based system with polycapillary optics to significantly improve the accuracy of quantitative phase retrieval.