I think we had this discussion on TGR, but I'll chime in here as well.
It's tough to generalize because river waves are so different. Broadly speaking though, the taller and greener the wave, the closer an appropriate board will look to an ocean-surfing shortboard. The flatter and mushier the wave, the more volume you're going to need to surf it. If you think about it, most of the cutbacks and aggressive carving on an ocean wave happens during the short-lived, "critical" period of the wave breaking, just a second or fractions of a second before the wave breaks into whitewater. By and large, river waves don't have that critical moment, since the waves are static/continuously breaking. So it takes a rare set of circumstances to produce a steep green face on a river without a breaking wave. You tend to see it at a small number of natural features and in artificial whitewater parks like Bend or RRP.
The more garden-variety river waves tend to be more of a wave/hole, with some manner of a pile on them. For those, you usually need more volume, since the wave has less height and steepness to accelerate you to a planing speed, and you need the volume so you don't sink straight through the aerated pile and have your tail dragging you off the wave. On the smaller mushier waves, you're not going to be getting big ocean-style cutbacks regardless of what kind of a board you're riding, but you can get long rides that are limited only by your skill/stamina.
This bit of your question: "Who's doing it and on what dimensions board AND on mostly natural river features- not cement created flumes where the water is moving 30 mph." It's happening at Skook and at the Lochsa Pipeline and at Glenwood on big days, all on ocean boards or river boards that look a lot like ocean boards with a little more width. But the conditions necessary for a wave steep enough, green enough, and tall enough are so rare that they're most often found in the concrete flumes.
I've written more on river surfing designs here
(all posts); here
(boards specifically), and here
(collecting a list of shapers specifically shaping river boards).