In Lurie's "A Survey of Elliptic Cohomology", he writes on page 14 that if $A$ is an ordinary commutative ring considered as an $E_\infty$-ring, then $A$-module spectra are the same thing as objects of the derived category of $A$-modules. This is mysterious to me. On the one hand, to an $A$-module spectrum $M$ we might associate the $A$-modules $\pi_n(M)$, but I don't know of any interesting maps between these; perhaps this will just end up being the homology of any representative chain complex of $A$-modules. But then, I certainly don't see a natural way of getting from an object of $\mathcal{D}(\mbox{Mod}_A)$ to an $A$-module spectrum.

Incidentally, what does this induce on the level of categories? The obvious first guess is that $A$-module spectra actually form a topological category and that passing to $\mathcal{D}(\mbox{Mod}_A)$ applies $\pi_0$.