Since G is finite, there is no problem with just repeating the proof in the 
case $G=e$, using $Z$-graded homotopy group functors on the orbit category. Take
$D^{\leq n}$ to be the spectra whose homotopy groups $\pi_q(X^H)$ are zero for $q>n$, 
and dually for $D^{\geq n}$.  The intersection for $n\leq 0$ and $n\geq 0$ consists 
of the Eilenberg-MacLane $G$-spectra $K(M,0)$ for Mackey functors $M$.   

Peter May