My comments a few lines below. I thought it might be worth mentioning that Dan's hexagonal representation of the torus appears "naturally" in the context of chessboard complexes. The chessboard complex corresponding to a chessboard of size m x n is a simplicial complex with one vertex for each square of the chessboard. A set of squares forms a face of the complex if and only if no two squares in the set appear in the same row or column. (Equivalently, rooks positioned on the given squares are pairwise non-attacking.) For example, if we label the columns with letters and the rows with integers, we have that the set {a1, b3, c2} forms a face, whereas {a1, b1, c2} and {a1, a3, c2} do not. In the case of a 4x3 chessboard with the four columns indexed by the letters A, B, C and D and the three rows indexed by the colors red, blue and yellow, we may represent the complex geometrically in the following manner. http://www.math.kth.se/~jakobj/pics/M43.jpg (Each small triangle in the picture, filled or not, belongs to the complex.) As the picture indicates, one may view this complex as a triangulation of a hexagonal region with opposite sides identified in the manner described by Dan:
Gareth's solution is correct, of course, and essentially the same as mine, which looks a little more symmetrical then the brick arrangement.
Namely, consider the tiling of the plane by regular hexagons. The group G of translations preserving this tiling must be a lattice in the plane. Hence the quotient of the plane by the lattice must be a torus, topologically.
Call the torus T.
Let one of the hexagons be called H. Since every point of the plane is equivalent to some point of H, we can obtain the same quotient torus T by just identifying the points of H that are equivalent by the group G of translations.
This means: Take the (closed) hexagon H and identify opposite edges in the simplest way -- identify each point of each edge with its closest poiwwnt on the opposite edge. This identification turns H into the torus T.
Some references for the chessboard complex can be found here: http://mathoverflow.net/questions/36791/is-the-4x5-chessboard-complex-a-link... Jakob