The author states a theorem on the additivity of the Euler characteristic -- or more generally, a suitably defined trace -- when applied to distinguished triangles of suitably dualizable objects in a refined notion of triangulated category. The main point of the paper is to write down the axioms needed to make such a theorem true, and to provide examples. The axioms, which seems to be complicated, are satisfied whenever the triangulated category arises as the homotopy category of a stable model category in the sense of \textit{M. Hovey} [``Model categories'', Math. Surv. Monogr. 63, Am. Math. Soc. (1999; Zbl 0909.55001), Chapter 7]. Most, if not all, important examples of triangulated categories arise under this rubric.