We calculate the annihilation decay rates of the ^3D_2(2^--) and ^3D_3(3^--) charmonia and bottomonia by using the instantaneous Bethe-Salpeter method. The wave functions of states with quantum numbers J^PC=2^-- and 3^-- are constructed. By solving the corresponding instantaneous Bethe-Salpeter equations, we obtain the mass spectra and wave functions of the quarkonia. The annihilation amplitude is written within Mandelstam formalism and the relativistic corrections are taken into account

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... y. This is important, especially for high excited states, since their relativistic corrections are large. The results for the 3g channel are as follows: Γ_^3D_2(cc̅)→ ggg = 9.24 keV, Γ_^3D_3(cc̅)→ ggg=25.0 keV, Γ_^3D_2(bb̅)→ ggg= 1.87 keV, and Γ_^3D_3(bb̅)→ ggg= 0.815 keV.