This paper presents a novel sampling-based planner which guarantees robustness for linear systems subject to bounded process noise, localization error, and/or uncertain environmental constraints. The proposed algorithm extends RRT*, efficiently generating and optimizing robust, dynamically feasible trajectories. During planning, state constraints are individually tightened for robustness against future uncertainty, while the input constraints can be tightened in order to apply feedback policies within planning for reduced conservatism. Simulation results demonstrate identification of smooth, guaranteed-safe trajectories in complex scenarios subject to both internal and external uncertainty, including cases where the uncertainty may be asymmetric and/or non-convex.