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Conducting high-aspect-ratio rods with 1-10 nm-scale diameters dispersed in poorly conducting matrices at extremely low, O(1%), volume fractions induce dramatic gains in bulk conductivity at rod percolation threshold. Experimentally [Nan, Shen, and Ma, Annu. Rev. Mater. Res., 40 (2010), pp. 131-151], bulk conductivity abandons the prepercolation, linear scaling with volume fraction that follows from homogenization theory [Zheng et al., Adv. Funct. Mater., 15 (2005), pp. 627-638], and then postpercolation jumps orders of magnitude to approach that of the pure rod macromolecular phase as predicted by classical percolation theory [Stauffer and Aharony, Introduction to Percolation Theory, CRC Press, Boca Raton, FL, 1994]. Our aim here is to use the orientational probability distribution functions from kinetic Brownian rod dispersion flow codes [Forest, Wang, and Zhou, Rheol. Acta, 44 (2004), pp. 80-93] to generate physical three-dimensional (3D) nanorod dispersions, followed by graph-theoretic algorithms applied to each realization to address two practical materials science questions that lie beyond the above theoretical results. How does bulk conductivity scale in the presence of anisotropy induced by shear film flow at and above rod percolation threshold? What are the statistical distributions of current within the rod phase? Our techniques reveal a robust exponential, large current tail of the current distribution above percolation threshold that persists over a wide range of shear rates and volume fractions; the exponential rates are spatially anisotropic, with different scaling in the flow, flow gradient, and vorticity axes of the film. The second moment of the computed current distributions furthermore captures and reproduces the bulk conductivity scaling seen experimentally. These results extend the scaling behavior for the classical setting of 3D lattice bond percolation [Shi et al., Multiscale Model. Simul., 11 (2013), pp. 1298-1310] to physical 3D nanorod dispersions with random centers of mass and shear-induced anisotropy in the rod orientational distribution. [ABSTRACT FROM AUTHOR]