Exact functors on the homotopy category

Question

Let $F: \mathcal{A} \to \mathcal{B}$ be an additive functor between abelian categories. When is the induced functor $K(F): K(\mathcal{A}) \to K(\mathcal{B})$ between the homotopy categories exact (as a functor of triangulated categories)? $K(F)$ always preserves shifts (by definition), and since the distinguished triangles in $K(\mathcal{A})$ are those homotopic to $A^\bullet \to B^\bullet \to \mathrm{Cone}(f) \to A^\bullet[1]$ for some map of complexes $f: A^\bullet \to B^\bullet$, I imagine that the answer is that $K(F)$ is exact if and only if it preserves cones (up to homotopy). But when is this the case? It seems to me that it should always be the case, for every $F$... In particular, is it the case that $K(F)$ is exact if $F$ is left or right exact?