Mixed Motives and Motivic Birational Covers

We introduce a tower of localizing subcategories in Voevodsky's big (closed under infinite coproducts) triangulated category of motives. We show that the tower induces an interesting finite filtration on the motivic cohomology groups of smooth schemes over a perfect field. With rational coefficients, this finite filtration satisfies several of the properties of the still conjectural Bloch-Beilinson-Murre filtration.