Let $\Omega$ be the standard volume on your Riemannian manifold, and $\phi$ a smooth function on M. A quick computation shows that $e^\phi \Omega$ is invariant by f if and only if the following cohomological equation is satisfied: $$ \phi(f^{-1}(x))-\phi(x)=log\ Jf(x)$$ where Jf is the jacobian of f. This implies for example that $Jf^n(x)=1$ for all $x\in Fix(f^n)$.
This later condition is in fact sufficient for C2 transitive Anosov diffeomorphisms (see e.g. Katok-Hasselblatt th 19.2.7). For these diffeos, this is also equivalent to saying that the SRB measure for f and the SRB measure for the inverse of f are equal (this is interesting because transitive Anosov diffeos always admit SRB measures, but of course not always smooth invariant measures).
In general one cannot expect that a smooth f-invariant measure to be given by a riemannian metric. Yet it is possible to find a smooth conjuguate of f that preserves any given riemannian volume: this is the Moser trick.
Let M be a compact riemannian manifold, $\Omega_0$ and $\Omega_1$ two volume forms with the same volume. Then there is a diffeo g such that $g^*\Omega_0=\Omega_1$.
The Moser theorem can also used to build a diffeo that does not preserve a Riemannian volume. Take an ergodic diffeo that preserves such a volume, and send that volume on a volume form that does not come from a riemannian metric.