A new momentum impulse relaxation method for obtaining the shear viscosity of Newtonian fluids using molecular dynamics simulations is introduced. The method involves the resolution of a decaying coarse-grain Gaussian velocity profile in a properly thermostated simulation box. This localized velocity profile, along with a modification of the periodic boundary conditions, allows computations in a periodic box with minimal phonon feedback due to periodicity. The short-time decay of the small-amplitude velocity profile yields shear viscosities for atomic and molecular species that are in quantitative agreement with those obtained using conventional techniques, but with more than an order of magnitude reduction in computational effort.