The flow of a viscous, incompressible MHD dusty Jeffrey fluid contained between two coaxial circular cylinders when the cylinders perform longitudinal oscillations along their co-axis is investigated. Three cases (i) when the inner cylinder oscillates while the outer cylinder moves with uniform velocity, (ii) when the outer cylinder oscillates while the inner cylinder moves with uniform velocity and (iii) when both the cylinders oscillate, are discussed. The velocity fields for the fluid and dust particles are obtained and the skin friction drag acting on each cylinder is determined. In three cases, it is observed that the fluid and dust velocities decrease with an increase in Hartmann number M, time and increase with increase in Jeffrey parameter λ1. The drag decreases near the inner cylinder and increases near the outer cylinder for different M and t. The opposite phenomena is observed for different λ1 and σ.