Question: I need to find the minimum cut on a graph. I've been reading about flow networks, but all I can find are maximum flow algorithms such as Ford-Fulkerson, push-relabel, etc. Given the max flow-min cut theorem, is it possible to use one of those algorithms to find the minimum cut on a graph using a maximum flow algorithm? How?

The best information I have found so far is that if I find "saturated" edges i.e. edges where flow equals capacity, those edges correspond to the minimum cut. Is that true? It doesn't sound 100% right to me. It is true that all edges on the minimum cut will be saturated, but I believe there might also be saturated edges which are out of the minimum cut "path".

Answer: 
From the source vertex, do a depth-first search along edges in the residual network (i.e., non-saturated edges and back edges of edges that have flow), and mark all vertices that can be reached this way. The cut consists of all edges that go from a marked to an unmarked vertex. Clearly, those edges are saturated and thus were not traversed. As you noted, there might be other saturated edges that are not part of the minimum cut.
I think this is what other people are saying, but I found it unclear so here's my explanation:

From the source node, do a flood fill of the graph, travelling only along edges with residual capacity, marking each visited vertex. You can use a DFS for this. Recall that back edges from a vertex have a residual capacity - equal to the flow along the forward edge (ie. r(u, v) = remaining capacity for edge (u, v), r(v, u) = flow(u, v)).

In effect, this determines the S part of the S-T cut of the graph.

The minimum cut will now be the set of edges such that one vertex is marked from your flood fill above, and the other vertex is not marked. These will be edges without residual capacity (otherwise you would have traversed them in your DFS), and together form the minimum cut.

After removing these edges, the set of unmarked vertices will form the T section of the cut.