Erik and leon k. were the first to inform me that the Australian grunge/alternative band Blueline Medic has recorded a version of Tori's Precious Things. The band has also performed this song before in concert. You can read about the recording on the band's official web site at bluelinemedic.com:
New BLM release
With "The Apology Wars" album almost due for a first birthday party, we're proud to announce the release of new Blueline Medic material. Early in June, Redline Records, the charming crew in Perth who are the band's Australian label, will unleash "New.Old.Live.", a split EP featuring three tracks from four bands: Blueline Medic, Americans Midtown and The Movielife and Sydney group Secondbest. Blueline's contributions include a studio recording of "Precious Things", the Tori Amos cover anyone who has been to show will be familiar with, and a live JJJ recording of a new track, "They'll Let You Know". As the release date nears we'll have more news of the split EP, gigs to launch it and some prizes you could win."
I am not sure if this EP is out yet, although it seems that it is not as of the time this page was last updated. The song has been heard on radio station Triple J in Australia.