This review considers three general classes of physical as opposed to phenomenological models of the shear elasticity of tissues. The first is simple viscoelasticity. This model has a special role in elastography because it is the language in which experimental and clinical data are communicated. The second class of models involves acoustic relaxation, in which the medium contains inner time-dependent systems that are driven through the external bulk medium. Hysteresis, the phenomenon characterizing the third class of models, involves losses that are related to strain rather than time rate of change of strain. In contrast to the vast efforts given to tissue characterization through their bulk moduli over the last half-century, similar research using low-frequency shear data is in its infancy. Rather than a neat summary of existing facts, this essay is a framework for hypothesis generation—guessing what physical mechanisms give tissues their shear properties.