This paper provides a general procedure for pricing American- and European-style interest rate derivatives within multifactor affine term structure models. We use a general dynamic programming formulation of the pricing equation and solve it by means of multilinear interpolations. This approximation relies to the maximum extent on the true conditional transition density of the state vector. As an illustration, we investigate the pricing of European interest rate swaptions, which cannot be analytically evaluated, as well as Eurodollar futures options. Using nine affine models, we compare our approach to the standard as well as the least-squares Monte Carlo simulation techniques. The results demonstrate that our approximation is accurate and converges rapidly. We also demonstrate that our approach remains well-behaved and outperforms the standard and least-squares Monte Carlo simulation methods for pricing deeply out-of-the money derivatives.