A number of groups have employed radio observations of the Galactic center to derive stringent constraints on the annihilation cross section of weakly interacting dark matter. In this paper, we show that electron energy losses in this region are likely to be dominated by inverse Compton scattering on the interstellar radiation field, rather than by synchrotron, considerably relaxing the constraints on the dark matter annihilation cross section compared to previous works. Strong convective winds, which are well motivated by recent observations, may also significantly weaken synchrotron constraints. After taking these factors into account, we find that radio constraints on annihilating dark matter are orders of magnitude less stringent than previously reported, and are generally weaker than those derived from current gamma-ray observations.