EDIT: As Misha and Deane pointed out, the question is not terribly well posed. So I will interpret it somewhat broadly to better reflect our actual understanding of canonical metrics on 3-manifolds and the Ricci flow.
By Perelman's proof of Thurston's geometrization conjecture, you can decompose any closed 3-manifold into geometric pieces. There are 8 types of geometric pieces, and they behave very simply under Ricci flow (they move just by scaling, etc). So the geometrization gives you essentially the class of metrics that you are asking for, with the constraint that you have to decompose your manifold into pieces first.
I don't think that there is a natural class of metrics, without decomposing the manifold into pieces first. In any case, it seems unlikely that the existence of such a class of metrics would yield to a simplification of Perelman's proof: E.g. a posteriori we know that every simply connected closed 3-manifold admits a positive Einstein metric. But of course, proving that means exactly proving the Poincare conjecture (as Perelman did).