  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
properly. This is important, especially for high excited states, since their
relativistic corrections are large. The results for the $3g$ channel are as
follows: $\Gamma_{^3D_2(c\bar c)\rightarrow ggg} = 9.24$ keV,
$\Gamma_{^3D_3(c\bar c)\rightarrow ggg}=25.0$ keV, $\Gamma_{^3D_2(b\bar
b)\rightarrow ggg}= 1.87$ keV, and $\Gamma_{^3D_3(b\bar b)\rightarrow ggg}=
0.815$ keV.
