A new method for extracting quantitative information from phase-contrast x-ray images obtained with microfocus x-ray sources is presented. The proposed technique allows rapid noninvasive characterization of the internal structure of thick optically opaque organic samples. The method does not generally involve any sample preparation and does not need any x-ray optical elements (such as monochromators, zone plates, or interferometers). As a consequence, samples can be imaged in vivo or in vitro, and the images are free from optical aberrations. While alternative techniques of x-ray phase-contrast imaging usually require expensive synchrotron radiation sources, our method can be implemented with conventional, albeit microfocus, x-ray tubes, which greatly enhances its practicality. In the present work, we develop the theoretical framework, perform numerical simulations, and present the first experimental results, demonstrating the viability of the proposed approach. We believe that this method should find wide-ranging applications in clinical radiology and medical research.