Discrete approximations to hyperbolic partial differential equations governing frictionless two-dimensional fluid flow are developed in Cartesian geometry for use over arbitrary triangular grids. A class of schemes is developed that conserves mass, momentum, and total energy. The terms of the governing equations are also approximated individually and their truncation error is examined. For test integrations, the schemes are applied to an equilateral triangular (homogeneous) grid on a beta plane. In one case, the same scheme is integrated over a square grid for comparison between four- and six-point differences. Both second- and fourth-order schemes are integrated and compared with a fine resolution solution.