Traditionally, the dry fly is fished by casting upstream over a specific rising trout. The fly is then dead-drifted, completely drag free, over the fish with no movement other than what the current provides. In this way the dry fly is to float past / over the trout. An exact imitation of the natural insects the trout has been eating truly helps the angler here to fool the fish. This was the method popularized & standardized in England 100+ years ago (on their smooth, gentle chalk streams) by Frederick Halford. This method came to America & became the most fished method of the dry fly here, too.
Cast down & across, or across, & put an upstream mend in your line during the forward cast (curve cast) to reduce drag. Allow your fly to drift the few feet it will without drag. Then, as your fly begins to drag &/or approaches a trout (or a probable place a trout might be if you're prospecting) give the line a very slight pull upstream. You only want to move the fly an inch or less. This motion gives you another foot or so of drag free drift & it draws the trout's attention to your fly as a living, moving object.
Of course, to use this method you'll need a fly that can stay afloat after you give it the "sudden inch" of movement - as Wright calls it. In the book he gives various recipes for flies to accomplish this.
He also delves deeper, showing how blind fishing, or prospecting, with a dry fly can be the most successful method for most of the season - especially in the low, slow waters of summer.
All in all this book is one of the best ever written on the subject of dry fly fishing. It was published in 1972 & I'm sure that there are folks fishing the dry fly downstream who never heard of Wright, but he was the pioneer. Done correctly, the "sudden inch" can be deadly. If you haven't read this book - & especially if you want to get better at fishing dry flies (as I do - I need all the help I can get), do yourself a favor & pick up a copy. I would go so far as to say that any fly fishing library is incomplete without a copy of "Fishing The Dry Fly As A Living Insect" on it's shelves.