  We study Grothedieck groups of triangulated categories using weight
structures, weight complexes, and the corresponding pure (co)homological
functors. We prove some general statements on $K_0$ of weighted categories and
apply it to Voevodsky motives endowed with so-called Chow weight structures. We
obtain certain "motivic substitutes" for smooth compactifications of smooth
varieties over arbitrary perfect fields; this enables us to make certain
unramified cohomology and Euler characteristic calculations that are closely
related to results of T. Ekedahl and B. Kahn.
