The Yarkovsky effect may play a key role in the orbital evolution of asteroids and near-Earth objects. To evaluate the acceleration under a wide range of conditions, a three-dimensional finite-difference solution to the heat equation is applied to homogeneous, spherical stony bodies with 1-, 10-, and 100-m diameters. This approach employs neither the linearized boundary conditions, the plane-parallel heat flow approximation, nor the assumption of fast rotation used in earlier work. Thus we can explore a wide range of orbital elements and physical properties. Our work agrees well with earlier results in the regimes where their approximations are valid. We investigate a wide range of spin states, including both the "seasonal" (very fast rotation) and "diurnal" (zero obliquity) extremes of the Yarkovsky effect. We find that, for orbits with high eccentricity, the semimajor axis can change much faster than for circular orbits. For such orbits, the orientation of the rotation axis with respect to the direction of pericenter is critical in determining the evolution. A stony main-belt asteroid of diameter 1 m on a high-eccentricity orbit could change its semimajor axis by more than 1 AU in 1.5 Myr.