Robots excel at avoiding obstacles but struggle to traverse complex 3-D terrain with cluttered large obstacles. By contrast, insects like cockroaches excel at doing so. Recent research in our lab elucidated how locomotor transitions emerge from locomotor-environment interaction for diverse locomotor challenges abstracted from complex 3-D terrain and the strategies to overcome them. Here we built on these fundamental insights to develop a cockroach-inspired legged robot, Omni-Roach, that integrated these strategies to achieve multi-modal locomotion and provide a robophysical model to study the tradeoff between multi-functionality and performance. The robot was based on the RHex design with six compliant legs and featured a rounded body with two wings that can open and a tail with pitch and yaw degrees of freedom. After two development and testing iterations, our robot was capable of overcoming all loco-motor challenges with a high performance and success rate. It traversed cluttered rigid pillars only 1.1× robot body width apart, a 2.5× hip height bump, a 0.75× body length gap, densely cluttered flexible beams only 65% body width apart, and self-righted within 4 seconds. Systematic beam traversal experiments further revealed that a downward-pointing tail oscillating laterally helps roll the body into beam gaps and break frictional and interlocking contact to traverse. Our work highlights the usefulness of multi-functional appendages and exaptation for large obstacle traversal.