This is clearly false, just consider the cylinder
$$ R_t \times S_{\theta} $$
with the product metric
$$g_\alpha=dt^2+\alpha^2 d\theta^2.$$
This is a flat metric so $Ric_{g_\alpha} = 0$. On the other hand, for $r>>\alpha$, it is easy to see $Area(\partial B_r)<8\pi \alpha$. Since $\alpha$ is arbitrary there is no uniform lower bound.
Maybe you need a uniform lower bound on the injectivity radius? (I'm not an expert on comparison geometry so don't know off the top of my head if this would suffice) [Edit: Or maybe this can only happen if the metric splits off an isometric euclidean factor].
[As an aside I can't seem to get math blackboard fonts to work anyone else have a problem with this?]