All good advice and right on the money. "Swinging" is actually how most spinning rod bait guys fish if you are familiar with their methods. Yep, just cast straight out, tighten the line, and let the current do its thing.
Adjust weight to get it to the part of the water column you want. With the belly and the drag, it will tend to lift your fly in the water column, thus it tends to require more weight than would be required to hit the same part of the water column with a dead drift.
Once you get the hang of it, its really easy to combine something close to a dead drift and the swing. Just cast upstream and mend until the fly gets in front of you, then stop mending and allow the swing. This is very effective as the fly will go from deep and start "swimming" towards the surface, imitating an emerging caddis which is what most wets are designed to imitate. The fish seem to respond to a fly coming off the bottom and going up very well, and they tend to hit as its rising off the bottom at the early stages of the swing. For this reason, I try to get my weight so that the fly bounces bottom for just a brief moment before it starts its ascent.