In this study, the results of a direct numerical simulation (DNS) of turbulent drag reduction by microfibers in a plane channel flow at a shear Reynolds number of Re = 950 are reported. For this purpose, we make use of a numerical solution of three-dimensional, time-dependent Navier-Stokes equations for the incompressible turbulent flow of a non-Newtonian fluid. The non-Newtonian stress tensor which is required to solve the problem depends on the orientation distribution of the suspended fibers, which is computed by a recently-proposed algebraic closure model. It is shown that the use of this algebraic closure, due to the great reduction in computational efforts, enables us to perform a DNS at high Reynolds numbers. Ultimately, statistical quantities of turbulence (in particular, the mean velocity profile, Reynolds stresses, etc.) are presented and discussed. Variations in the isotropy of the Reynolds stress tensor are explained by the aid of Lumley anisotropy map.