The velocity of ultrasound longitudinal waves (speed of sound) is emerging as a valuable biomarker for a wide range of diseases, including musculoskeletal disorders. Muscles are fiber-rich tissues that exhibit anisotropic behavior, meaning that velocities vary with the wave-propagation direction. Therefore, quantifying anisotropy is essential to improve velocity estimates while providing a new metric related to muscle composition and architecture. For the first time, this work presents a method to estimate speed-of-sound anisotropy in transversely isotropic tissues using pulse-echo ultrasound. We assume elliptical anisotropy and consider an experimental setup with a flat reflector parallel to the linear probe, with the muscle in between. This setup allows us to measure first-arrival reflection traveltimes using multistatic operation. Unknown muscle parameters are the orientation angle of the anisotropy symmetry axis and the velocities along and across this axis. We derive analytical expressions for the nonlinear relationship between traveltimes and anisotropy parameters, including reflector inclinations. These equations are exact for homogeneous media and are useful to estimate the effective average anisotropy in muscles. To analyze the structure of this forward problem, we formulate the inversion statistically using the Bayesian framework. We demonstrate that anisotropy parameters can be uniquely constrained by combining traveltimes from different reflector inclinations. Numerical results from wide-ranging acquisition and anisotropy properties show that uncertainties in velocity estimates are substantially lower than expected velocity differences in the muscle. Thus, our approach could provide meaningful muscle anisotropy estimates in future clinical applications.