Just to add my .02 cents here. It's not so much what your wrist is/is not doing, but where you're rod tip is traveling. A form follows function sort of thing. If you're bringing your rod straight up and down on your forward cast the line, and the fly, will be coming directly at your rod tip.
Cant the rod tip over a bit until it passes your ear; at that point the line/fly are already committed to follow their original plan (slightly off to the side).
The above is also directly related to casting a 2-hander. If you 'wack' your rod with your fly, count on the rod coming through in too vertical a position. (Let's leave the wind out of this