In this paper, we present a design methodology for the generation of magnetohydrodynamic (MHD) complex flows in a simple straight microchannel inside which electrodes are patterned on the two side walls and the bottom wall. The Lorentz force, a driving force for the MHD flow, can be variously induced in the microchannel by changing the applied voltages at the patterned electrodes and the magnetic field. The design methodology for determining the required condition for the applied voltage at each electrode is discussed for generating axial, transverse, sinusoidal, and multi-vortical flows under a given magnetic field. Three-dimensional CFD (computational fluid dynamics) simulations demonstrate the successful generation of the designed complex flows in a microchannel. The mixing performances of axial, sinusoidal, and multi-vortical flows were evaluated as an application example of the present MHD complex flows. The present design methodology may be also applied for the precise control of the fluid flow inside a microchannel, not to mention the micromixing.