It is shown that the random walk simulation technique, through the choice of a suitable set of rules, can be adapted to the problem of solving the linearized Poisson-Boltzmann equation, for general cases involving constant surface potentials. In order to validate the technique, the results of the method are compared to the exact solutions for a number of analytically tractable problems. Excellent agreements between the two sets of results are found in all cases. Further examples of the application of the method are given by considering systems involving nonideal colloidal particles. In particular, the solutions of the linearized Poisson-Boltzmann equation for simple models of porous colloids, as well as particles with nonuniform surface potentials, have been obtained. The implication of these preliminary results for the electrostatic interactions between such nonideal colloidal particles are briefly discussed.