As part of a program to identify the physical conditions in the jets of γ-ray-flaring blazars detected by Fermi, including the role of shocks in the production of high-energy flaring, we obtained 4 years of 3-frequency, centimeterband total flux density and linear polarization monitoring observations of the radio-bright blazar S5 0716+714 with the University of Michigan 26-m paraboloid. Light curves constructed from these data exhibit a series of rapid, high-amplitude, centimeter-band total flux density outbursts, and changes in the linear polarization consistent with the passage of shocks during the γ-ray flaring. The observed spectral evolution of the radio-band flares, in combination with radiative transfer simulations incorporating propagating shocks, was used to constrain the shock and jet flow conditions in the parsec-scale regions of the jet. Eight forward-moving, transverse shocks with unusually-strong shock compression factors, a very fast Lorentz factor of the shocks of 77, a bulk Lorentz factor of the flow of 20, a viewing angle of 12 • , and an intrinsic opening angle of the radio jet of 5.2 • were identified.