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We demonstrate the waveguiding of Lamb waves in a locally resonant phononic crystal (LRPC) and we present an analysis of the guiding of elastic waves in straight and bent waveguides. The finite element method combined with the supercell technique was used to analyze the band gap and the dispersion relation of LRPC waveguides. Unlike the traditional phononic crystals, we show the possibility of guiding only one confined mode inside a LRPC waveguide. We discuss the confinement and the transmission of the guided mode as a function of the width of the waveguide based on both the band structure and the displacement field.