There isn't a descent theory for derived categories per se - one can't glue objects in the derived category of a cover together to define an object in the base. (Trying to apply the usual Barr-Beck to the underlying plain category doesn't help.)

But I think the right answer to your question is to use an enriched version of triangulated categories (differential graded or $A_\infty$ or stable $\infty$-categories), for which there is a beautiful Barr-Beck and descent theory, due to Jacob Lurie. (This is discussed at length in the n-lab I believe, and came up recently on the n-category cafe (where I wrote basically the same comment here..)
This is proved in DAG II: Noncommutative algebra. In the comonadic form it goes like this. Given an adjunction between $\infty$-categories (let's call the functors pullback and pushforward, to mimic descent), if we have

- pullback is conservative (it respects isomorphisms), and
- pullback respects certain limits (namely totalizations of cosimplicial objects,
which are split after pullback)

then the $\infty$-category downstairs is equivalent to comodules over the comonad
(pullback of pushforward). (There's an opposite monadic form as well)
This can be verified in the usual settings where you expect descent to hold.
In other words if you think of derived categories as being refined to $\infty$-categories (which have the derived category as their homotopy category), then everything you might want to hold does.

So while derived categories don't form a sheaf (stack), their refinements do:
you can recover a complex (up to quasiisomorphism) from a collection of complexes on a cover, identification on overlaps, coherences on double overlaps, coherences of coherences on triple overlaps etc.
More formally: define a sheaf as a presheaf $F$ which has the property that
for an open cover $U\to X$, defining a Cech simplicial object $U_\bullet=\{U\times_X U\times_X U\cdots\times_X U\}$, then $F(X)$ is the totalization of the cosimplicial object $F(U_\bullet)$. Then enhanced derived categories form sheaves (in appropriate topologies) as you would expect. This is of course essential to having a good theory of noncommutative algebraic geometry!