We propose a randomized implementation of stochastic model predictive control (MPC). As a proxy for the expected cost, which may not be efficiently computable, the algorithm minimizes the empirical average cost under N random samples of the uncertain influences on the system. The setting is an imperfectly-observed linear system with multiplicative and additive uncertainty, convex, deterministic control constraints, and convex costs that may include penalties on state constraint violations. In this setting, each sample-average MPC subproblem is a feasible convex program that, under mild regularity conditions, yields consistent estimators of the stochastic MPC subproblem's optimal value and minimizers. Under stronger assumptions, the full sample-average MPC control trajectory is asymptotically optimal for stochastic MPC as N → ∞. A numerical example shows that even for small N, sample-average MPC can significantly improve performance relative to certainty-equivalent MPC.