The present study investigates the serious effects of rotation of material axes on the free dynamic response of composite plates or panels with "Bonded Double Doubler Mint Systems". The "Plate Adherends" and the "Upper and Lower Doubler Plates" are connected through the relatively very thin adhesive layers. The "Bonded Double Doubler Joint System" is considered in taans of the "System. 1" and the "System.2". In the "System. 1", the material directions of "Plate Adherends" are rotated 90(0) (about z-axis) while there is no change in the material axes of the "Double Doubler Plates". In the "System.2", the material directions of the "Double Doubler Plates" are rotated 90(0) (about z-axis), while there is no change in the material axes of the "Plate Adherends". All plate elemnts of the "System. 1" and the "System.2" are assumed to be dissimilar "Orthotropic Mindlin Plates" with the transverse shear deformations and the transverse (or bending) moments of inertia and the rotary moments of inertia. The upper and lower adhesive layers are linearly elastic continua with dissimilar material properties and with unequal thicknesses. The damping effects in. all plate elements and also in adhesive layers are neglected. The entire theoretical analysis for both "Systems.1 and 2" is based on the "Orthotropic Mindlin Plate Theory". For this purpose, the dynamic equations of the left and the right "Plate adherends" and of the "Upper and Lower Doubler Plates" and the equations of the adhesive layers are combined to-gather with the stress resultant - displacement expressions of the plate elements. Then, after some algebric manipulations and combinations, and with the "Classical Levy's Solutions" the original dynamic equations are finally reduced into the two new sets of the "Governing System of the First Order O.D.E's" in compact matrix forms with the "state vectors" for the "System. 1" and "System.2", respectively. In this way, the original "Initial and Boundary Value Problem" (or the free vibrations problem) is converted to the "Multi - Point Boundary Value Problem" of Mechanics and Physiscs. In the case of both "Systems.1 and 2", these results facilitate the direct application of the present method of solution that is the "Modified Transfer Matrix Method (MTMM) (with Interpolation Polynomials)". The aforementioned "Governing Equations" for both "Systems.1 and 2" are numerically integreted by making use of the "(MTMM) (with Interpolation Polynomials)". Thus, the natural frequencies and the mode shapes of the "Systems. 1" and the "System.2" are graphically presented for the same "Support Conditions". The comparison of the numerical results corresponding to each "System. 1" and "System.2" for the same "Support Conditions" is considered leading to some very important conclusions.