A novel unidirectional freeze casting method is used to fabricate graphene aerogels (GAs) having a highly aligned porous structure. The high orientation of graphene sheets in the as-fabricated GAs results from the expulsion of graphene oxide (GO) sheets by rapidly advancing ice fronts, forcing them to gather between the aligned ice crystals. The resulting unidirectional GAs (UGAs) possess ultralow densities with high porosities, large surface areas and excellent electrical conductivities after reduction. Solid UGA/epoxy composites are fabricated by vacuum-assisted infiltration of liquid epoxy into the pores of UGA, followed by curing. The composites exhibit an extremely low percolation threshold of 0.007 vol% due to the 3D interconnected, aligned conductive networks of UGA. This value, to the authors' best knowledge, is among the lowest percolation thresholds for all graphene/polymer composites reported in the open literature. In addition, significant anisotropic electrical conductivities of the composites are observed, which is potentially useful for many niche applications. An improved analytical model is developed based on the interparticle distance concept to predict the percolation behavior of the current UGA/epoxy composites. Critical factors that control the percolation threshold of nanocomposites are identified, including the orientation states and the aspect ratio of fillers.