A new approximate solution for the velocity profile of steady incompressible magnetohydrodynamic (MHD) flows in a rectangular microchannel driven by the Lorentz force is proposed. Mean velocity and mass flow rate in a channel, subsequently derived, can be used efficiently for many MHD-based microfluidic applications, including the design of a MHD-based microfluidic network without resorting to costly full-scale computational fluid dynamics. The closed-form solutions, provided for both direct-current (dc) and alternating-current (ac) electric and magnetic fields, are in simple forms, without any series or functions to evaluate, and so can be readily used for inverse or control problems associated with MHD-based lab-on-a-chip (LOC) devices. Extensive comparisons with previous analytical, computational, and experimental results are performed, and summarized in the present study. The proposed solutions are shown to agree better with existing experimental and computational reports than previous approximations and are to be used in a broad range of MHD-based LOC applications with both dc and ac fields with required accuracy.