Create a few new desktops in Mission Control if you don't already have them. (Launch Mission Control, then move your cursor to the top-right corner of the screen and click in the big +.)
Go to a desktop where there is an open application window.
Drag that window to the edge of the screen, and the screen will slide to the next desktop. You can keep dragging it through your desktops if you keep it against the edge; it will shift one desktop per second.
When you've reached the desktop where you want to put that window, release it and position it as you want. Note that this won't cycle through to the beginning or end of your desktops. In other words, if you have three desktops, and move a window to the right-most desktop, you can't bring it back to the first one by moving it to the right again.
[kirkmc adds: I had never noticed this, because I invoke Mission Control using a hot corner, and move windows like that. This is similar to the way you move app icons on iOS devices]