Bourginion showed that if a compact manifold does not admit positive scalar curvature metrics, then any scalar flat metric is Ricci-flat; I suppose this is what you mean when you write "strongly scalar flat". But in three dimensions, the Ricci curvature determines the full curvature tensor, in particular, a Ricci-flat metric is flat. So any non-flat three-manifold which does not admit positive scalar curvature will provide an example. 

If I'm not mistaken, Gromov and Lawson proved that a compact three-manifold admits positive scalar curvature if and only if its prime decomposition contains no aspherical factors; note, this was before the Poincaré conjecture had been verified, so there would have been a caveat at the time of publication.

So $T^3\#S^1\times S^2$ is an example of a compact three-manifold of type (C).