We propose a novel method for calculating resonances in three-body systems with repulsive Coulomb interactions. The method is based on the solution of a set of Faddeev and Lippmann-Schwinger integral equations. The resonances of the three-body system are defined as the complex-energy solutions of the homogeneous Faddeev integral equations. We show how the kernels of the integral equations should be continued analytically in order to get the resonances. As a numerical illustration a model for the three-α system is solved.