Many animals, modern aircraft, and underwater vehicles use fusiform, streamlined body shapes that reduce fluid dynamic drag to achieve fast and effective locomotion in air and water. Similarly, numerous small terrestrial animals move through cluttered terrain where three-dimensional, multi-component obstacles like grass, shrubs, vines, and leaf litter also resist motion, but it is unknown whether their body shape plays a major role in traversal. Few ground vehicles or terrestrial robots have used body shape to more effectively traverse environments such as cluttered terrain. Here, we challenged forest-floor-dwelling discoid cockroaches (Blaberus discoidalis) possessing a thin, rounded body to traverse tall, narrowly spaced, vertical, grass-like compliant beams. Animals displayed high traversal performance (79 ± 12% probability and 3.4 ± 0.7 s time). Although we observed diverse obstacle traversal strategies, cockroaches primarily (48 ± 9% probability) used a novel roll maneuver, a form of natural parkour, allowing them to rapidly traverse obstacle gaps narrower than half their body width (2.0 ± 0.5 s traversal time). Reduction of body roundness by addition of artificial shells nearly inhibited roll maneuvers and decreased traversal performance. Inspired by this discovery, we added a thin, rounded exoskeletal shell to a legged robot with a nearly cuboidal body, common to many existing terrestrial robots. Without adding sensory feedback or changing the open-loop control, the rounded shell enabled the robot to traverse beam obstacles with gaps narrower than shell width via body roll. Such terradynamically 'streamlined' shapes can reduce terrain resistance and enhance traversability by assisting effective body reorientation via distributed mechanical feedback. Our findings highlight the need to consider body shape to improve robot mobility in real-world terrain often filled with clutter, and to develop better locomotor-ground contact models to understand interaction with 3D, multi-component terrain.