This paper considers the pricing of Bermuda-style swaptions in the LIBOR market model and its extensions. Application of lattice methods to this model class is generally not feasible because of the large number of state variables, so instead a simple technique to incorporate early exercise features into the Monte Carlo method is considered. The approach here involves a direct search for an early exercise boundary parametrized in intrinsic value and the values of still-alive swaptions. The results of the proposed algorithm are compared against prices obtained from Markov chain approximations, nonrecombining trees, and finite difference methods. The proposed algorithm is fast and robust, and produces a lower bound on Bermudan swaption prices that appears to be very tight for many realistic structures. The paper contains several numerical results against which other methods can be tested.