I've flown with camera gear, 95 percent of the time with my Betacam, and never had a problem.
Just make sure you have a battery with you so you can turn it on and off, open the tape door, if they want to check it out but bottom line it's not an issue.
It's considered your carry on bag.
Don't expect to take hand carry the camera and another bag onto the plane though.
Always keep in mind that your checked baggage may not find it way to you, so make sure you have what you need to shoot with. I always make sure I have a couple of batteries, charger, and media (before boxes of tapes) This way you only have to worry about buying clothes.
I fly all the time with cameras, almost every week. Never called ahead. I always try to carry on tape stock and enough batteries to shoot for half day in case the rest of the gear gets delayed or lost, that saved my rear more than once.
If you find yourself flying in a very small plain there is a big chance your camera bag doesn't fit in the overhead, be ready to take the camera out of the case and share your seat wit it.
We flew from Seattle to Copenhagen 9/06 with a lay-over and missed connection in Newark in between and still somehow managed to have all of our stuff arrive with us (I'm still not sure how that happened).
But we carried on the essentials, most expensive or most fragile in a single Pelican - camera, wireless mics, shotgun mic, batteries, etc., and other items such as firewire drive, tripod head, tapes, XLR cables, etc. in a carry-on backpack. Light kit and tripod sticks were checked.
The only problem was having the backpack filled with cables, tapes and drive (and some socks and underwear) unpacked by security at every point, and then having to pack it all back in.
The light kit was packed in a very large Pelican case and was a few pounds overweight. We reported this in Seattle and paid the fee, but forgot in Copenhagen when coming back and paid nothing. It may, for certain things, pay to feign ignorance (but not an otherwise endorsed piece of advice.)
When we go back in December we're flying non-stop and will invest in a travel hand cart.