Many boundary value problems arising in materials science modeling involve complicated boundary shapes and boundary data, making analytic solution based on conventional differential equation methods difficult. In particular, it is important to develop effective computational methods for calculating the transport properties of polymers and complex-shaped particle aggregates arising in materials science and biology. As a first step towards attacking this broad class of problems, we focus on the problem of calculating basic solution transport properties of isolated particles having essentially any geometry using a novel computational method involving path integration. The basic concepts behind the method are described and the method is validated in cases where exact analytic, or at least highly accurate numerical estimates, are known for comparison. After defining our method, validating its accuracy, some applications of the program are given to some non-trivial problems; nanotubes treated as either rigid rods or ensembles worm-like chains of finite cross-section, DNA, nanoparticles with grafted chain layers and knotted polymers. The path-integration method is evidently a powerful tool for computing basic transport properties of complex-shaped objects and should find wide application in polymer science, nanotechnological applications and biology.