While it commonly assumed that Wii cannot play DVDs because of some sort of hardware complication, a group of hackers have actually proved that Wii can play DVD films, and have released a boot disc to let you do just that.
The Wii DVD playback software comes in the form of a CD boot disc (you download and burn the software to a CD yourself). Inserting this CD into a Wii fitted with a mod-chip boots the Wii into a menu. Then you eject the CD, insert your DVD movie and the Wii springs into action as a fully capable DVD player.
Modding your Wii to play DVDs isn't advised (all sorts can go wrong) but this is good news because it proves that the only thing stopping your little white box from running DVD films is the lack of supporting software.
This throws up the possibility of someone like Datel, who made the import game-playing Freeloader on GameCube entirely without support or endorsement from Nintendo, may be able to make a boot disc that plays DVDs on normal un-modded Wii.
And why wouldn't Nintendo release such a disc? Or at least a dongle like you needed with the Xbox. We're awaiting a reply on that very question.
Meanwhile, if you really want to watch movies on Wii you can download or purchase PC applications, such as X-OOM Software's X-OOM Movies on Wii, that allows you to rip DVD films onto an SD memory stick (via PC) for playback on Wii in the Photo Channel. Playback quality, however, is questionable.