We present an effective technique for suppressing the vortex-induced vibrations of bluff bodies by eliminating the von Kármán street formed in their wake. Specifically, we find that small amounts of combined windward suction and leeward blowing around the body modify the wake instability and lead to suppression of the fluctuating lift force. Three-dimensional simulations and stability analysis are employed to quantify our findings for the flow past fixed and flexibly mounted circular cylinders.