The flows induced by acoustic streaming in a three-dimensional side-heated parallelepiped cavity of length Ax representative of crystal growth configurations are numerically studied. Both the structure of the flows and their stability properties are determined. The flows have different symmetries, belonging to the group D4 for pure streaming, Z2xZ2 for pure buoyancy, and Z2 for the mixed case, but these symmetries are generally broken at the first bifurcation points. Bifurcation diagrams are obtained which show that the flows become oscillatory periodic at a Hopf bifurcation, either directly on the primary steady solution branch, or on a secondary branch which bifurcates from the primary branch at a steady bifurcation point. The critical Grashof numbers for these bifurcation points are calculated as a function of the cavity length Ax, the Prandtl number Pr and the acoustic streaming parameter A. The thresholds are generally found to increase when the acoustic streaming contribution is enhanced, which indicates a stabilizing effect induced by acoustic streaming and may explain the observed improvement of the crystal quality when ultrasound waves are applied during the growth process. Destabilization effects are, however, found in some parameter range.