We have studied the Ostwald ripening of three-dimensional islands on a homogeneous surface with an original off-lattice kinetic Monte Carlo algorithm. In this algorithm, adatom trajectories are highly simplified, while still ensuring that the adatom fluxes between islands are exactly described. From the simulations, we obtained the evolution of the island size distribution over a large time range. The simulations obtained are compared with the results of numerical integration of rate equations derived from a mean-field approximation. Both results indicate that the equilibrium radius of the islands follows a power-law behavior in the limit of a very dilute phase, with an exponent close to 1/4. A general, excellent agreement is obtained, showing the validity of our approach, whereas the validity of the mean-field approximation is discussed for a very small mean island size, or for a large fraction of the surface covered by the islands.