It doesn't take long when fishing in a creek, stream or river to feel that you've exhausted a given area - especially if you're targeting only the areas most likely to hold trout. When it's time to move, you know you have to be sneaky - trout have keen eyesight and we've constantly discussed on this site just how skittish they can be. You basically have two directions to go - upstream or downstream. Most experts recommend moving upstream, but there are arguments in favor of both directions.
The reason you so often hear anglers recommending moving upstream as you fish is because the fish are facing upstream. Since trout have sharp eyes, you have less chance of spooking them by sneaking up behind them.
On the other hand it's more difficult to move upstream and you're more likely to lose your footing as you fight against the current - which could be dangerous in water waist deep or higher.
If you choose to move downstream, the trout will see you quicker than if you move upstream, since they will be facing towards you. This only comes into play, though in narrow creeks and you should be able to travel pretty effectively and stealthily downstream in larger streams and rivers.
Another advantage of moving downstream is that you'll move quieter. Fighting the current can create a lot of splashing and disturb nearby fish.
Really the decision of how to move depends on where you're going. That is, you want to move towards the fish. Take into consideration the places trout are most likely to be hiding. Are they upstream of where you are, or downstream? Will you have to walk further in one direction or the other to get to the next productive hole?
Take into account of who's around you. If you must walk past another fisherman, take a wide berth to make sure not to get in the way of his back cast and not disturb any fish. A better idea, is to walk away from any nearby anglers. They have very likely either caught or spooked the fish in the hole they're working. You're much better off being the first person to fish a new hole than to try to catch another angler's leftovers.
Also, consider the sun. If the sun is upstream, you won't want to move downstream, since you will cast a shadow making you very easy to spot by wary trout. You also won't want to move upstream with the sun behind you, even though the effect is reduced by the fish facing away from you - they will still notice when you block out the sun.
Your movement in the stream ultimately shouldn't be dictated by an arbitrary rule. Take into account your surroundings and where the fish are most likely to be biting and move accordingly. Of course, if you're in a hole that's producing fish, don't move - you never leave fish for fish.