Magnetorheological elastomer (MRE) is a solid-state smart material with field-dependant dynamic shear modulus. But, its lower shear modulus, which is about 0.388MPa, does prevent its application. Sandwich configuration is an alternative to apply MRE in engineering since the outer thin skins will strengthen the bulk flexural stiffness and the transverse flexibility of the MRE core will affect the bulk flexural dynamic performance. In this paper, the field-dependant dynamic property of MRE-based sandwich beams, composed of conductive skins or non-conductive skins, is addressed theoretically through a high order model. By defining the maximum field-induced relative change of the bulk fexural dynamic stiffness as controllability index, structure designs to yield maximum controllability are presented through a non-dimensional analysis. The simulation on simply supported MRE-based sandwich beam indicates: (1) the anti-resonant frequencies and resonant frequencies of the sandwich beam change with applied magnetic fields up to 30%; (2) the bulk field-dependant flexural dynamic property is mainly depended on the field-dependant shear modulus of the MRE core; and, (3) there is an optimal combination of the thickness of the core and the thickness of the skins for maximum controllability; (4) around the optimal combination point, the controllability/mass ratio can be enhanced dramatically though decreasing the core thickness; (5) the normalized density of the skins affects the controllability slightly when the Young's modulus of the skins is low. This work indicates that sandwich structures can well utilize the controllable property of MRE to realize applicable stiffness changeable devices.