In order to understand the underlying mechanisms of animals' agility, dexterity and efficiency in motor control, there has been an increasing interest in the study of gait patterns in biological and artificial legged systems. This paper presents a novel approach to the study of gait patterns which makes use of intrinsic mechanical dynamics of robotic systems. Each of these robots consists of a U-shape elastic beam and exploits free vibration to generate different gait patterns. We developed a conceptual model for these robots, and through simulation and real-world experiments, we show three distinct mechanisms for generating four different gait patterns in these robots.