Finding all edges of an undirected graph which are in some cycle in linear time
It is not difficult to prove that an edge is a cut-edge if and only if it does not lie on a cycle. So you can find the set $E'$ of cut edges, and what you are then looking for is $E \setminus E'$.

For finding $E'$, there are several well-known linear-time algorithms. Wikipedia gives e.g., Tarjan's bridge-finding algorithm.