We study the stability of two-dimensional gravity-capillary periodic water waves in the case of large surface tension. In this parameter regime, predictions based on model equations suggest that periodic traveling waves are stable with respect to two-dimensional perturbations, and unstable with respect to three-dimensional perturbations which are periodic in the direction transverse to the direction of propagation. In this talk we show that these predictions are confirmed for the full Euler equations describing this hydrodynamic problem.