The present investigation deals with the study of plane waves and the fundamental solution in rotating modified couple stress generalized thermoelastic solid with two-temperatures. The heat conduction equation with the fractional-order derivative and the formulation of the problem is applied in the context of the three-phase-lag model (3PHL), Green–Naghdi theory (G-N III) with energy dissipation and Green–Naghdi theory (G-N II) without energy dissipation. The medium is assumed initially stressed and normal mode analysis will be used to obtain the general solution for any set of boundary conditions. Some comparisons have been shown in figures to estimate the effects of the fractional-order, the hall current, rotation, and modified stress on all the studied fields.