This paper presents an approach that integrates the geometric notion of clearance (distance to the closest obstacle) into sampling-based motion planning to enable a robot to safely navigate in challenging environments. To reach the goal destination, the robot must obey geometric and differential constraints that arise from the underlying motion dynamics and the characteristics of the environment. To produce safe paths, the proposed approach expands a motion tree of collision-free and dynamically feasible motions while maintaining locally maximal clearance. In distinction from related work, rather than explicitly constructing the medial axis, the proposed approach imposes a grid or a triangular tessellation over the free space and uses the clearance information to construct a weighted graph where edges that connect regions with low clearance have high cost. Minimum-cost paths over this graph produce high-clearance routes that tend to follow the medial axis without requiring its explicit construction. A key aspect of the proposed approach is a route-following component which efficiently expands the motion tree to closely follow such high-clearance routes. When expansion along the current route becomes difficult, edges in the tessellation are penalized in order to promote motion-tree expansions along alternative high-clearance routes to the goal. Experiments using vehicle models with second-order dynamics demonstrate that the robot is able to successfully navigate in complex environments. Comparisons to the state-of-the-art show computational speedups of one or more orders of magnitude.