Summary: This paper deals with the analytic continuation of periodic orbits of conservative dynamical systems with three degrees of freedom. For variations of any parameter (or integral), we use a predictor-corrector algorithm to compute the initial conditions for periodic orbits pertaining to the family. The method proposed here is not restricted to symmetric problems and, since the procedure involves the computation of variational equations, a side effect is the trivial computation of the linear stability of periodic orbits. As an illustration of the robustness of the method, we compute several families of periodic orbits of the restricted three-body problem.