We present a study of the decays B+-->rho+gamma, B0-->rho0gamma, and B0-->omegagamma. The analysis is based on data containing 347 x 10(6) BB[over ] events recorded with the BABAR detector at the PEP-II asymmetric B factory. We measure the branching fractions B(B+-->rho+)gamma)=(1.10_(-0.33)(+0.37)+/-0.09)x10(-6) and B(B0-->rho0gamma)=(0.79(-0.20)(+0.22)+/-0.06)x10(-6), and set a 90% C.L. upper limit B(B0-->omegagamma)<0.78 x 10(-6). We also measure the isospin-averaged branching fraction B(B-->(rho/omega)gamma)=(1.25(-0.24)(+0.25)+/-0.09)x10(-6), from which we determine |Vtd/Vts|=0.200(-0.020)(+0.021)+/-0.015, where the first uncertainty is experimental and the second is theoretical.