  We construct small amplitude gravity-capillary water waves with small nonzero
vorticity, in three spatial dimensions, bifurcating from uniform flows. The
waves are symmetric, and periodic in both horizontal coordinates. The proof is
inspired by Lortz' construction of magnetohydrostatic equilibria in
reflection-symmetric toroidal domains. It relies on a global representation of
the vorticity as the cross product of two gradients, and on prescribing a
functional relationship between the Bernoulli function and the orbital period
of the water particles. The presence of the free surface introduces significant
new challenges. In particular, the resulting free boundary problem is not
elliptic, and the involved maps incur a loss of regularity under Fr\'echet
differentiation. Nevertheless, we show that a version of the
Crandall-Rabinowitz local bifurcation method applies, by carefully tracking the
loss of regularity.
