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This paper introduces a model for the propagation of multiple types of waves in a compressed granular chain placed on an elastic V-shaped rail. The system possesses translational, rotational, and torsional degrees of freedom, and the different waves interact due to the presence of Hertzian contacts between adjacent spheres and between the spheres and the rail. Additional tangential force and torque between the spheres and the rail governed by the Hertz–Deresiewicz law are accounted for. This allows an exclusive tuning of the shear and torsional stiffness coefficients and leads to a complex coupled wave dynamics. Tunable dispersion relations including zero group velocity mode, accidental degeneracy and mode hybridisation are investigated in the linear regime. It is shown that the tangential force-dependent shear stiffness between the spheres and the rail strongly influences the dispersion relations, while the influence of the torque-dependent torsional stiffness is much weaker. The inclination angle of the rail is examined as another tunable degree of freedom, which modifies the wave dispersion not only by changing the shear coupling but also by influencing the coupling among different types of waves. The tunability of the wave dynamics in the granular chain evidenced in this work offers a flexible way of designing acoustic systems for selective vibration filtering and wave manipulation. • Tunable wave dispersion including zero group velocity mode and accidental degeneracy for a granular chain–rail structure. • An additional tuning mechanism due to the coupling of the V-shaped rail and the granular chain. • Model accounts for the effects of additional tangential force and torque using the Hertz–Deresiewicz contact law. • Wave transmission in finite systems preserves dispersion features of the infinite chain. [ABSTRACT FROM AUTHOR]