The gravity-capillary water-wave problem concerns the irrotational flow of a perfect fluid in a domain bounded below by a rigid bottom and above by a free surface under the influence of gravity and surface tension. In the case of large surface tension the system has a family of traveling two-dimensional periodic waves for which the free surface has a periodic profile in the direction of propagation and is homogeneous in the transverse direction. We show that these periodic waves are linearly unstable under spatially inhomogeneous perturbations which are periodic in the direction transverse to propagation. As a consequence, the periodic waves undergo a dimension-breaking bifurcation generating a family of spatially three-dimensional solutions which are periodic in both the direction of propagation and the transverse direction.