The mechanics of the actin cytoskeleton regulates cell morphogenesis during essential physiological processes. However, the spatially heterogeneous and dynamic nature of the actin cytoskeleton make mechanical measurements and modeling challenging. Here we develop a new approach to probe the mechanics of the contractile actin cytoskeleton by integrating optogenetic control of RhoA, live cell imaging and traction force microscopy. Local activation of RhoA stimulates local contraction, leading to increased traction forces that rapidly propagate across the cell via stress fibers and drive actin flow towards the region of heightened RhoA. Surprisingly, the flow reverses direction when local RhoA activation stops. These experimental data are used to constrain a physical model, which demonstrates that stress fibers are elastic-like, even at time scales exceeding turnover of constituent proteins. We identify zyxin as a regulator of stress fiber mechanics, as they are fluid-like in its absence. Such molecular control of actin mechanics likely plays critical roles in regulation of morphogenic events.