The tortuosities of fibrous media in the heretofore unexplored transition and ordinary regimes are computed using a Monte Carlo scheme based on the Einstein equation for random walkers. The model structure is that of fully penetrable cylinders (FPC) in a unit simulation volume. The mean square displacement technique is combined with the first passage time distribution to accelerate the progress of the walkers at low Knudsen number. The results include the computation of transition regime transport coefficients for the first time. The calculated ordinary tortuosities are approximately equal to the reciprocal of the porosity over a wide range, while the transition tortuosities are shown to deviate from the reciprocal porosity with a simple dependence on Knudsen number. The limits of the transition regime are shown to correspond roughly to Knudsen numbers of 0.50 and 100, respectively. The calculated Knudsen tortuosities are shown to improve on earlier results obtained by the authors using a flux-based technique.