This work was supported primarily by the MRSEC program of the National Science Foundation under DMR-0079996. The applicability of standard lattice percolation models to a random 2-D grain structure is explored. A random network based on the triangle lattice is proposed as a more appropriate model, and results in a higher percolation threshold (0.711 compared with 0.653 for the standard hexagonal lattice). The triple junction constraint inherent in grain boundary structures is subsequently applied to the new network. This results in a lowering of the percolation threshold to 0.686; this is opposite to its effect on the standard hexagonal lattice. The effect of varying the network's "grain shape" distribution on the percolation threshold is also considered.