We report the investigation of surface acoustic waves (SAWs) propagating on a 1D phononic surface (PS) by the heterodyne-detected transient reflecting grating technique. A suitable experimental configuration enables the excitation of traveling SAWs with variable wave numbers and the measurement of their propagation with temporal and spatial resolution. Using the full characterization of the band diagram of this PS, as reported previously (Malfanti I., Taschin A., Bartolini P., Bonello B. and Torre R., J. Mech. Phys. Solids, 59 (2011) 2370), we studied the dispersion properties of the SAWs as the wave number approaches the Brillouin zone edge. SAW-packet group velocities show a clear slowing-down process approaching the BZ edge, with a measurable minimum velocity of about 140 m/s