Robotic systems need to be able to plan control actions that are robust to the inherent uncertainty in the real world. This uncertainty arises due to uncertain state estimation, disturbances, and modeling errors, as well as stochastic mode transitions such as component failures. Chance-constrained control takes into account uncertainty to ensure that the probability of failure, due to collision with obstacles, for example, is below a given threshold. In this paper, we present a novel method for chance-constrained predictive stochastic control of dynamic systems. The method approximates the distribution of the system state using a finite number of particles. By expressing these particles in terms of the control variables, we are able to approximate the original stochastic control problem as a deterministic one; furthermore, the approximation becomes exact as the number of particles tends to infinity. This method applies to arbitrary noise distributions, and for systems with linear or jump Markov linear dynamics, we show that the approximate problem can be solved using efficient mixed-integer linear-programming techniques. We also introduce an important weighting extension that enables the method to deal with low-probability mode transitions such as failures. We demonstrate in simulation that the new method is able to control an aircraft in turbulence and can control a ground vehicle while being robust to brake failures.