This paper presents a novel strategy for designing passively adaptive, statically stable walking robots with full body mobility that are exactly constrained and non-redundantly actuated during stance. In general, fully mobile legged robots include a large number of actuated joints, giving them a wide range of controllable foot placements but resulting in overconstraint during stance, requiring kinematic redundancy and redundant control for effective locomotion. The proposed design strategy allows for the elimination of actuation redundancy, thus greatly reducing the weight and complexity of the legged robots obtained and allowing for simpler control schemes. Moreover, the underconstrained nature of the resulting robots during swing allows for passive adaptability to rough terrain without large contact forces. The strategy uses kinematic mobility analysis tools to synthesize leg topologies, underactuated robotics design approaches to effectively distribute actuation constraints, and elastic elements to influence nominal leg behavior. Several examples of legged robot designs using the suggested approach are thoroughly discussed and a proof-of-concept of a non-redundant walking robot is presented.