Significance: A major challenge in the physical and life sciences is to see the invisible by quantifying the structure of materials and living tissues below nominal imaging resolution. Measurement of molecular diffusion performed at a macroscopic level is a common noninvasive structural probe. Interpreting its results involves identifying which structural features, out of a myriad of parameters, most strongly affect the diffusion coefficient. Here we address this ill-posed problem by classifying media into a relatively small number of universality classes, based on the nature of long-range structural correlations. We show that the time dependence of a macroscopic diffusion coefficient distinguishes between these classes. Using this approach, we identify dominant restrictions to MRI-measured water diffusion in muscles and in cortical gray matter.