I take my fishing rod with me all the time while backpacking. Very few things are better than fresh fish instead of re-hydrated meat while on a trip. I especially like bringing a bit of butter, lemon juice, and pepper in a bottle to fry the fish in. Keeps just fine over several days.
As for bears, I read a report from Parks Canada that said people fishing are one of the most likely groups to have a bear encounter. However, that usually happens to people fishing up and down streams where the bear can't hear or smell them. As I understand it, near water is the most common place to have a bear encounter.
To minimize the risk I try to fish from a relatively stationary position with some decent sight lines, always carry my bear spray, and I bring an odor proof bag to carry the fish in if I need to carry it to camp. I like to cook the fish either the campground eating area (more people = more safety) or away from both the water and camp if not at an official campground.
What kind of rod were you planning to bring? I have found a collapsible spinning rod with a couple of Black Fury lures to be dynamite with the trout in high alpine Rockies lakes.
Not the fastest, but I get to where I am going and enjoy it.