In this note, we show that for a large class of nonlinear wave equations with odd nonlinearities, any globally defined odd solution which is small in the energy space decays to 0 in the local energy norm. In particular, this result shows nonexistence of small, odd breathers for some classical nonlinear Klein Gordon equations, such as the sine-Gordon equation and $\phi ^4$ and $\phi ^6$ models. It also partially answers a question of Soffer and Weinstein (Invent Math 136(1): 9–74, p 19 1999) about nonexistence of breathers for the cubic NLKG in dimension one.