In this paper we present the "random walk on the boundary" method for the rapid solution of integral equations that arise in electrostatics and related areas. This method is a Monte Carlo method based on the construction of a Markov chain that is readily interpreted as a random walk along the boundary over which integration in the integral equation is taken. To illustrate the usefulness of this technique, we apply it to the computation of the capacitance of the unit cube. Obtaining the capacitance of the cube usually requires computing the charge density, and this problem has been used as a benchmark by many in the field for algorithms of this kind. Here, the "random walk on the boundary" method does not require charge density computation, and obtains the capacitance of the cube within a statistical error of 2.7×10 -7, the most accurate estimate to date.