We consider solving triangular systems of linear equations on a hypercube multiprocessor. Specifically, we propose a fast parallel algorithm, applicable when the triangular matrix is distributed around the cube by column in a wrap fashion. Numerical experiments indicate that the new algorithm is very efficient. A theoretical analysis confirms that the total running time varies linearly, with respect to the matrix order, up to a threshold value of the matrix order, after which the dependence is quadratic. Moreover, we show that total message traffic is essentially the minimum possible. Finally, we describe an analogous row-oriented algorithm.