Homebrewer Erant has created a program that allows the Wii to read DVDs. It comes bundled with a media player program that lets you play DVD movies as well as view video files stored on an SD card, without any hardware modifications. You still need to be able to run homebrew on your Wii, but that's done easily enough.
The commenters in the announcement post seem to be having mixed success with the program, with results ranging from, at best "Excellent! Works like a charm." to "It doesn't seem to play DVDs on my Wii," to the worst possible outcome, "watching Hackers (1995) right now."
As a sort of proof of concept of the new DVD-reading capability for uses other than video, a patch will soon be available for Nintendo 64 emulator Wii64 to allow ROMs to be read from DVDs.