We present the results of a search for gravitational-wave bursts associated with 137 gamma-ray bursts (GRBs) that were detected by satellite-based gamma-ray experiments during the fifth LIGO science run and first Virgo science run. The data used in this analysis were collected from 2005 November 4 to 2007 October 1, and most of the GRB triggers were from the Swift satellite. The search uses a coherent network analysis method that takes into account the different locations and orientations of the interferometers at the three LIGO-Virgo sites. We find no evidence for gravitational-wave burst signals associated with this sample of GRBs. Using simulated short-duration (<1>s) waveforms, we set upper limits on the amplitude of gravitational waves associated with each GRB. We also place lower bounds on the distance to each GRB under the assumption of a fixed energy emission in gravitational waves, with typical limits of D ~ 15 Mpc (E_GW^iso / 0.01 M_o c^2)^1/2 for emission at frequencies around 150 Hz, where the LIGO-Virgo detector network has best sensitivity. We present astrophysical interpretations and implications of these results, and prospects for corresponding searches during future LIGO-Virgo runs.