A new method is introduced for simulating the motion of particles in a viscoelastic fluid. The viscoelastic fluid is represented as a suspension of finite-extension-non-linear-elastic (FENE) dumbbells in Newtonian solvent. Instead of using an averaging technique to derive a continuum constitutive model, such as that of Chilcott and Rallison, we calculate directly the particle-particle and particle-bead interactions by using a modified version of the Stokesian dynamics method. This method is applied to a series of sedimentation problems, including the sedimentation of a single sphere, a non-spherical particle, and two spheres. It is found that the drag of a single sphere can be increased significantly by the presence of the dumbbells, and at high Deborah numbers (De) the sphere velocity becomes time-periodic. Non-spherical particles rotate as they fail such that their long axis is ultimately pointed in the direction of gravity. Two sedimenting spheres are in most cases attracted to each other, and they turn such that their line-of-centers is in the direction of gravity. The rotational velocity of the spheres in a side-by-side configuration is in the same direction as it would be in Newtonian fluid, in keeping with experimental observations found in the literature.