A. Suslin and V. Voevodsky have realized a triangulated category which is supposed to be the bounded derived category of the hypothetical abelian category of mixed motives over , predicted by Grothendieck–Beilinson–Deligne. There are variants developed by Hanamura and M. Levine. There is a different “derived” approach to mixed motives, namely the -homotopy theory of F. Morel and V. Voevodsky.
See at motive the section Contructions of the derived category of mixed motives.
wikipedia article on motives and references therein.
A. Beilinson, V. Vologodsky, A DG guide to Voevodsky’s motives, math.AG/0604004
M.V. Bondarko, Differential graded motives: weight complex, weight filtrations and spectral sequences for realizations; Voevodsky vs. Hanamura, math.AG/0601713
M.V. Bondarko, Weight structures and motives; comotives, coniveau and Chow-weight spectral sequences: a survey, arxiv:0903.0091