Many important properties of a macromolecule can be expressed in terms of averages over the trajectories of diffusing particles that begin in the medium surrounding the molecule and terminate at its surface. These properties include its translational hydrodynamic friction coefficient and the Smoluchowski rate constant for diffusion-limited reactions. In this paper we introduce a first-passage algorithm (FPA) for calculating such quantities. This algorithm uses certain exact Green’s functions, or propagators, for the Laplace equation to eliminate the need to construct explicitly those portions of a diffusing particle’s trajectory that are not near an absorbing object. The algorithm is especially efficient for studying objects that contain large voids or have very irregular surfaces, such as macromolecules. Diffusion algorithms were previously shown to give accurate results for the quantities we study. In this paper, we show that first-passage methods make these algorithms more accurate and efficient. In future work, we expect to present systematic results for the properties of globular proteins.