In this article, a new mathematical model of magneto-thermoelasticity has been constructed to investigate the transient phenomena in the context of a new consideration of heat conduction with fractional order, where the anisotropic fibre-reinforced medium is rotating with an uniform angular velocity. The governing equations for Green-Naghdi theory have been formulated and solved analytically employing the normal mode analysis. Numerical results for the temperature, displacement and stresses are obtained and have been depicted graphically. According to the numerical results and corresponding graphical representations, conclusions about this new theory have been constructed. Finally, some comparisons have been shown to estimate the effect of magnetic field, reinforcement and the presence of non-local fractional parameter on all the studied fields.