Jack makes a good point about casting past the trout and dragging the fly into position, with slack in the line. You have to remember that it's how the fly is drifting when it gets to the fish that counts. Many times you will have a great drift that gets whipped by the current every time it reaches the trout, and this can be frustrating. Sometimes it seems they know just where to lie just to frustrate you. You don't have to drag the fly from upstream either. It can be dragged from across stream, or around an obstruction, etc. You may cast to a spot that's nowhere near the current seam you want, so you can drag it into the seam to a spot that will make it drift nicely over the trout's lie. Just be sure not to spook the fish by dragging your line across him.
As far as upstream, downstream, across stream, quartering, etc., any one of these may work best depending upon the situation you're in. And not just the current. You have to deal with casting obstructions, wind, wading obstructions, avoiding spooking the trout, and so on. Be prepared to use whatever method is needed for the situation, and don't limit yourself to any particular "flyfishing rules."