Dynamic stability allows running animals to maintain preferred speed during locomotion over rough terrain. It appears that rapid disturbance rejection is an emergent property of the mechanical system. In running robots, simple motor control seems to be effective in the negotiation of rough terrain when used in concert with a mechanical system that stabilises passively. Spring-like legs are a means for providing self-stabilising characteristics against external perturbations. In this paper, we show that a quadruped robot could be able to perform self-stable running behaviour in significantly broader ranges of forward speed and pitch rate with a suitable mechanical design, which is not limited to choosing legs spring stiffness only. The results presented here are derived by studying the stability of the passive dynamics of a quadruped robot running in the sagittal plane in a dimensionless context and might explain the success of simple, open loop running controllers on existing experimental quadruped robots. These can be summarised in (a) the self-stabilised behaviour of a quadruped robot for a particular gait is greatly related to the magnitude of its dimensionless body inertia, (b) the values of hip separation, normalised to rest leg length, and leg relative stiffness of a quadruped robot affect the stability of its motion and should be in inverse proportion to its dimensionless body inertia, and (c) the self-stable regime of quadruped running robots is enlarged at relatively high forward speeds. We anticipate the proposed guidelines to assist in the design of new, and modifications of existing, quadruped robots. As an example, specific design changes for the Scout II quadruped robot that might improve its performance are proposed.