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).

As pointed out by Deane, any volume form can be realised as the volume associated to a Riemannian metric. Embed your manifold M in R^n, extend your volume form $f dvol_{eucl}$ to a neighborhood of the manifold, then take the restriction of the metric $f^{2/m}g_{eucl}$ to M . In fact, it is even 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 $g^*\Omega_0=\Omega_1$.