A novel method of calculating viscosity from molecular dynamics simulations is developed, benchmarked, and tested. The technique is a transient method which has the potential to reduce CPU requirements for many conditions. An initial sinusoidal velocity profile is overlaid upon the peculiar velocities of the individual molecules in an equilibrated simulation. The transient relaxation of this initial velocity profile is then compared to the corresponding analytical solution of the momentum equation by adjusting the viscosity-related parameters in the constitutive equation that relate the shear rate to the stress tensor. The newly developed Transient Molecular Dynamics (TMD) method was tested for a Lennard-Jones (LJ) fluid over a wide range of densities and temperatures. The simulated values were compared to an analytical solution of the boundary value problem for a Newtonian fluid. The resultant viscosities agreed well with those published for Equilibrium Molecular Dynamics (EMD) simulations up to a dimensionless density of 0.7. Application of a linear viscoelastic Maxwell constitutive equation was required to achieve good agreement at dimensionless densities greater than 0.7. When the Newtonian model is used for densities in the range of 0.1 to 0.3 and the Maxwell model is used for densities higher than 0.3, the TMD method was able to predict viscosities with an uncertainty of 10% or better. Application of the TMD method to multi-site molecules required the Jeffreys constitutive equation to adequately fit the simulation responses. TMD simulations were performed on model fluids representing n-butane, isobutane, n-hexane, water, methanol, and hexanol. Molecules with strong hydrogen bonding and Coulombic interactions agreed well with NEMD simulated values and experimental values. Simulated viscosities for nonpolar and larger molecules agreed with NEMD simulations at low to moderate densities, but deviated from these values at higher densities. These deviations are explainable in terms of potential model inaccuracies and the shear-rate dependence of both NEMD and TMD viscosity values. Results show that accurate viscosity predictions can be made for multi-site molecules as long as the shear-rate dependence of the viscosity is not too large or is adequately addressed.