We extend the method of transition-path sampling to the case of deterministic dynamics. This method is a Monte Carlo procedure for sampling the ensemble of trajectories that carry a many-particle system from one set of stable or metastable states to another. It requires no preconceived notions of transition mechanisms or transition states. Rather, it is from the resulting set of suitably weighted dynamical transition paths that one identifies transition mechanisms, determines relevant transition states and calculates transition rate constants. In earlier work, transition-path sampling was considered in the context of stochastic dynamics. Here, the necessary modifications that make it applicable to deterministic dynamics are discussed and the modifications illustrated with microcanonical simulations of isomerization events in two-dimensional seven-atom Lennard-Jones clusters.