Quartersawn stock would be the least likely solid material to warp.

Otherwise, can you use a rail and stile to enclose a piece of 1/4" plywood?

At this point, I was thinking 3/4 plywood, 1/4 inch edge band, painted the same color as the walls. I really don't want the doors to be a "design feature," if that makes sense. I wasn't even going to use knobs. That's why I wasn't considering frame and panel, which would tend to highlight the geometry.

Perhaps the best thing is just to cut the doors in half, so a four foot tall cabinet has two doors? A 12x24 ply door, painted, would probably be ok.

MDF. Edgebanding with solid wood would be a good idea, so that the edges could stand more abuse.

Re: MDF

I have to laugh at myself for not thinking of this. When I was more active as a woodworker, it was standard practice to say this stuff should never, ever be used in a wet location, but I currently have some scrap that has been sitting outside my shop, in the rain and snow for three weeks, and it looks almost as good as new. I guess this makes sense, since it's 98 percent waterproof glue.

The tiny amount of humidity in my bath isn't going to ruin a well painted MDF cabinet door.

Excellent suggestion. Extra points for it being the cheapest option, but I do half to deduct a couple points because I will have to lift the 100 lb sheets onto my saw...

(considering 1/2 inch...)

Oh, now, don't be so dramatic; they're only 70 pounds or so. Go with the 3/4" thickness so you can use concealed hinges.