Algorithm for finding all the heaviest edges in a graph
In line with what Amir Nasr said in his comment above, I think the cycle property of MSTs would imply that finding a solution to your problem in linear time would find an MST in linear time.

From the Wikipedia page on MSTs and a few of the specific algorithms, it seems you can only currently do better be exploiting specific properties of your graph or using randomness.

You may find the $O(\left\vert{E}\right\vert\log{}\left\vert{V}\right\vert)$ complexity preferable to the $O(\left\vert{E}\right\vert+\left\vert{V}\right\vert\log{}\left\vert{V}\right\vert)$, and it seems that would just require a different choice of data structures or using either Kruskal's or Borůvka's.

For complexity with no $\log{}$ factors, it seems (for lack of a known general linear, deterministic, MST algorithm) you need either dense graphs, integer weights, or to use the expected linear time MST algorithm below.

https://en.wikipedia.org/wiki/Expected_linear_time_MST_algorithm