Stability of the nonlinear periodic solutions of the Benjamin-Ono equation with respect to one-dimensional perturbations and taking into account transverse modulations is analytically studied. The full set of eigenfrequencies and of small perturbations is found in the one-dimensional case. The dispersion equation for long-wave oblique perturbations is obtained. These perturbations are shown to be unstable under either sign of the dispersion of the linear dispersion relation. The instability limits and the maximum growth rates are determined in the extreme cases of strongly and weakly nonlinear waves.