I don't know why it took me so long to realize, but last night I found out about the Wii Transfer Tool, for free on Nintendo's eStore. My son wanted to play Pokemon Rumble, which I had downloaded for him on the Wii some time ago. While...

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Nintendo seems to be providing backwards compatibility one system behind. So the Wii U is compatable with the Wii. The Wii compatable with the Gamecube.
The Wii U doesn't have compatability with the GameCube and it's unfortunate it doesn't. But the Wii did have GameCube controller ports and yhey didn't add those to the Wii U.
Glad to hear transferring your stuff is trivial though. At least someone made it work.
Good point and sad to think it's not compatible with Gamecube discs (which seems true from a quick Google search). However, their virtual console has games from many of their previous consoles. So, in that way it's similar to how the Playstation 4 is going to be backwards compatible. The Wii U doesn't have a Gamecube memory slot, however I know a Gamecube controller accessory is due or out as they used it during some Smash Bros. videos. Still, it gives me a decent reason to keep my Wii. I still have my Gamecube and N64. Sadly, I don't know what happened to my SNES, but I do also have a NES. =) My son and I also have many generations of the Gameboy and DS.
I am going to keep my Wii around for quite some time too, it just makes me feel good to have it under the TV even though I don't play it so often.