Rheological properties of living cells play a key role in the control of cell shape, growth, movement, and contractility, yet little is known about how these properties are governed. Past approaches to understanding cell mechanics focused on the contributions of membranes, the viscous cytoplasm, and the individual filamentous biopolymers that are found within the cytoskeleton. In contrast, recent work has revealed that the dynamic mechanical behavior of cells depends on generic system properties, rather than on a single molecular property of the cell. In this paper, we show that a mathematical model of cell mechanics that depicts the intracellular cytoskeleton as a tensegrity structure composed of a prestressed network of interconnected microfilaments, microtubules, and intermediate filaments, and that has previously explained static cellular properties, also can predict fundamental dynamic behaviors of living cells.