Brown dwarfs and directly imaged giant planets exhibit significant evidence for active atmospheric circulation, which induces a large-scale patchiness in the cloud structure that evolves significantly over time, as evidenced by infrared light curves and Doppler maps. These observations raise critical questions about the fundamental nature of the circulation, its time variability, and the overall relationship to the circulation on Jupiter and Saturn. Jupiter and Saturn themselves exhibit numerous robust zonal (east-west) jet streams at the cloud level; moreover, both planets exhibit long-term stratospheric oscillations involving perturbations of zonal wind and temperature that propagate downward over time on timescales of ∼4 years (Jupiter) and ∼15 years (Saturn). These oscillations, dubbed the Quasi Quadrennial Oscillation (QQO) for Jupiter and the Semi-Annual Oscillation (SAO) on Saturn, are thought to be analogous to the Quasi-Biennial Oscillation (QBO) on Earth, which is driven by upward propagation of equatorial waves from the troposphere. To investigate these issues, we here present global, three-dimensional, high-resolution numerical simulations of the flow in the stratified atmosphere-overlying the convective interior-of brown dwarfs and Jupiter-like planets. The effect of interior convection is parameterized by inducing small-scale, randomly varying perturbations in the radiative-convective boundary at the base of the model. Radiative damping is represented using an idealized Newtonian cooling scheme. In the simulations, the convective perturbations generate atmospheric waves and turbulence that interact with the rotation to produce numerous zonal jets. Moreover, the equatorial stratosphere exhibits stacked eastward and westward jets that migrate downward over time, exactly as occurs in the terrestrial QBO, Jovian QQO, and Saturnian SAO. This is the first demonstration of a QBO-like phenomenon in 3D numerical simulations of a giant planet.