Extensive nonequilibrium molecular dynamics simulations have been carried out for liquid decane, hexadecane, and tetracosane at densities corresponding to atmospheric pressure and near ambient temperatures. The strain-rate-dependent viscosity has been obtained for strain rates ranging over several orders of magnitude. At high strain rate, the viscosities for all alkanes studied here have similar values and exhibit similar power-law shear-thinning behavior with a slope between about -0.40 and -0.33. Accompanying this shear thinning is the onset of orientational order and the alignment of the alkane molecules with the flow direction. The alignment angle tends to 45 degrees at very low strain rate and is significantly smaller at high strain rate, This suggests that the chains substantially align in the flow direction and that the dominant motion at high strain rate is the sliding of the chains parallel to the flow. At low strain rate, the shear viscosity shows a transition to Newtonian behavior. The Newtonian viscosity can be obtained from the plateau value of the shear viscosity at the lowest strain rates calculated from the nonequilibrium molecular dynamics simulation (NEMD). This is demonstrated by comparing the viscosity of decane obtained by extrapolating the NEMD simulation with an independent calculation using the standard Green-Kubo method. The transition from the non-Newtonian regime to the Newtonian regime is also correlated with the disappearance of orientational order and with the longest relaxation time of the liquid alkanes simulated.