This response is unfortunately not about simply-connected manifolds, I overlooked that assumption by the author of the question. So, this is not an answer. See my comment below.

Gromov writes [in this paper][1] (see Section 0.4) an example of an odd dimensional (not simply-connected) manifold with non-vanishing minimal volume, by constructing a lower bound involving the so-called **simplicial volume** (and a corresponding metric $g$).

[1]: http://www.ihes.fr/~gromov/PDF/4[35].pdf