The biomechanical behavior of tissues under mechanical stimulation is critically important to physiological function. We report a combined experimental and modeling study of bioengineered 3D smooth muscle microtissues that reveals a previously unappreciated interaction between active cell mechanics and the viscoplastic properties of the extracellular matrix. The microtissues' response to stretch/unstretch actuations, as probed by microcantilever force sensors, was dominated by cellular actomyosin dynamics. However, cell lysis revealed a viscoplastic response of the underlying model collagen/fibrin matrix. A model coupling Hill-type actomyosin dynamics with a plastic perfectly viscoplastic description of the matrix quantitatively accounts for the microtissue dynamics, including notably the cells' shielding of the matrix plasticity. Stretch measurements of single cells confirmed the active cell dynamics, and were well described by a single-cell version of our model. These results reveal the need for new focus on matrix plasticity and its interactions with active cell mechanics in describing tissue dynamics.