Learning to fight fish in fast water is a difficult thing to get your head round at times. Pull too hard and you'll break your tippet, don't pull hard enough and you wont have a chance. Here are a few words of advice.
1. Get downstream of your fish where possible. Pulling a fish up current in a fast river is a recipe for disaster, firstly with the combination of current and a running fish then you are highly likely to either bust the fish off your rip the hook out. Secondly at that angle you are pulling the fly right out of the fishes mouth (rather than back into it).
2. Keep the rod high. By holding your rod tip high you'll be able to keep as much line as possible out of the current and control the amount of pressure on the fish.
3. Work as a team. Yeah i know fly fishing is a great solitary sport but having your mate net your fish will greatly decerease the time spent playing your fish and enable you to work on keeping the right angles on the fish while your friend concentrates on getting the net under it.
4. When it runs down the rapids, make sure you chase it.