The superimposition of mechanical oscillations can lead to a remarkable modification in the friction factor (C-f) with a fixed flowrate for viscous non-Newtonian fluids in a pipe but cannot affect the flow of Newtonian types. Therefore, less driving pressure is required for fluids with a shear-thinning nature, while more for shear-thickening types. A validated computational fluid dynamics (CFD) model is employed to compare the efficacy of three modes of oscillation: longitudinal, transverse, and rotational within the laminar flow region. Rheologies considered here are power-law fluids and Bingham plastic types. At a given Reynolds number (Re), C-f is found to be a function of rheological parameters (i.e., flow behavior index, n and Bingham number, Bn) and the vibration intensity, lambda. Longitudinal oscillation is the most effective, with transverse oscillation also producing comparable results. Rotational oscillation causes the least effect on the viscosity and therefore, the friction factor is even nearly unchanged while lambda <= 20 for power-law fluids, and lambda <= 5 for Bingham plastic rheologies. Expressions for C-f are proposed, giving a quantitative estimation on the effect of mechanical vibration on the driving pressure.