In short, I have never run into trouble carrying on rod, reel, line, flies. BUT I have heard enough horror stories about line and saltwater flies that in the past couple of years, I have generally checked my saltwater flies. And sometimes, I have just packed my vest/fanny-pack with forceps/nippers in checked baggage. Never had an issue, but I have worried about it.
I also own three 5-piece rods (with this week's addition of an 8-weight) for the express purpose of packing them in luggage because I always fear running into issues (but I haven't ever run into issues with my 4-piece rods).