We study the generation of vorticity in the magnetic boundary layer of buoyant magnetic tubes and its consequences for the trajectory of magnetic structures rising in the solar convection zone. When the Reynolds number is well above 1, the wake trailing the tube sheds vortex rolls, producing a von Kármán vortex street, similar to the case of flows around rigid cylinders. The shedding of a vortex roll causes an imbalance of vorticity in the tube. The ensuing vortex force excites a transverse oscillation of the flux tube as a whole so that it follows a zigzag upward path instead of rising along a straight vertical line. In this paper, the physics of vorticity generation in the boundary layer is discussed and scaling laws for the relevant terms are presented. We then solve the two-dimensional magnetohydrodynamic equations numerically, measure the vorticity production, and show the formation of a vortex street and the consequent sinusoidal path of the magnetic flux tube. For high values of the plasma beta, the trajectory of the tubes is found to be independent of β but varying with the Reynolds number. The Strouhal number, which measures the frequency of vortex shedding, shows in our rising tubes only a weak dependence with the Reynolds numbers, a result also obtained in the rigid-tube laboratory experiments. In fact, the actual values measured in the latter are also close to those of our numerical calculations. As the Reynolds numbers are increased, the amplitude of the lift force grows and the trajectory becomes increasingly complicated. It is shown how a simple analytical equation (which includes buoyancy, drag, and vortex forces) can satisfactorily reproduce the computed trajectories. The different regimes of rise can be best understood in terms of a dimensionless parameter, χ, which measures the importance of the vortex force as compared with the buoyancy and drag forces. For χ2<<1, the rise is drag dominated and the trajectory is mainly vertical with a small lateral oscillation superposed. When χ becomes larger than 1, there is a transition toward a drag-free regime and epicycles are added to the trajectory.