Well GIMP can do simple 3D stuff, you can map a layer to a sphere, box, or cylinder. It's under Filters -> Map -> Map Object. Anything more complicated than that and you probably need to use something like Gmax. But since all I needed was a sphere and a cylinder it worked pretty well.
The sphere I just mapped the source image directly. For the flat Earth I first had to convert the source image to a polar projection map, which I did by using Filters -> Distorts -> Polar Coordinates.