We consider two equivariant equations admitting structurally stable heteroclinic cycles. These equations stem from mode equations for the Rayleigh-Benard convection and a model for turbulent layers in wall regions with riblets. Breaking the symmetry causes several different bifurcations to occur which can be explained by bifurcations of codimension two of homoclinic orbits for non-symmetric systems. In particular, stable periodic solutions of different symmetry type, other complicated heteroclinic cycles or geometric Lorenz attractors may emanate. Moreover, we develop stability criteria for the bifurcating periodic solutions. In general, their stability type differs from the stability properties of the original heteroclinic cycle.