Bipedal walking is one of the most important hallmarks of human that robots have been trying to mimic for many decades. Although previous control methodologies have achieved robot walking on some terrains, there is a need for a framework allowing stable and robust locomotion over a wide range of compliant surfaces. This work proposes a novel biomechanics-inspired controller that adjusts the stiffness of the legs in support for robust and dynamic bipedal locomotion over compliant terrains. First, the 3D Dual-SLIP model is extended to support for the first time locomotion over compliant surfaces with variable stiffness and damping parameters. Then, the proposed controller is compared to a Linear-Quadratic Regulator (LQR) controller, in terms of robustness on stepping on soft terrain. The LQR controller is shown to be robust only up to a moderate ground stiffness level of 174 kN/m, while it fails in lower stiffness levels. On the contrary, the proposed controller can produce stable gait in stiffness levels as low as 30 kN/m, which results in a vertical ground penetration of the leg that is deeper than 10% of its rest length. The proposed framework could advance the field of bipedal walking, by generating stable walking trajectories for a wide range of compliant terrains useful for the control of bipeds and humanoids, as well as by improving controllers for prosthetic devices with tunable stiffness.