We calculate the bulk viscosity of the massive pion gas within unitarized chiral perturbation theory. We obtain a low-temperature peak arising from explicit conformal breaking due to the pion mass and another peak near the critical temperature, dominated by the conformal anomaly through gluon condensate terms. The correlation between bulk viscosity and conformal breaking supports a recent QCD proposal. We discuss the role of resonances, heavier states, and large-N_(c) counting.