Last year, Wire magazine had a cover story called "New Weird America", which tried to document the latest "movement" in weird/psychedelic music. It's often called psychedelic folk or acid folk. Many of the groups are large/loose collectives, often very mysterious, and it's usually hard to find their stuff outside of mail-order or very independently-run record stores. Below, I've tried to compile stuff that Amazon has listed, such as Sticks and Stones May Break My Bones but Names Will Never Hurt Me or Liberation. Most of the music is very loose, with washed-out vocals/tribal chants, occasional horns, and blends of traditional/acoustic and electronic instruments.
Like most "new" music, much of this genre already existed in obscure folk, rock, or jazz outifts from the 60's and 70's. And with all "weird" music, the best source is usually Sun Ra. You could basically listen to any of his stuff from 1960-1977 to understand what I mean, but a solid example would be Atlantis. Although many other free jazz musicians inspired much of the new freakout music, I personally think that 1969-1974 era Miles Davis provided a unique rock-funk-jazz basis. Check out The Complete Jack Johnson Sessions to get an idea.
Some of the best hippie/tribal rock came out of Germany in the late 60's; especially from Amon Duul: Psychedelic Underground. Really obscure US 60's psychedlic folk (and not that great) was A Cid Symphony. For the true folk influence, the source is/was John Fahey: America, and maybe also the somewhat obscure Robbie Basho, Guitar Soli.
OK, on to more contemporary stuff. Above, I listed the No Neck Blues Band (NNCK: Intonomancy) and Jackie-O Mother f---er (JOMF: Fig. 5), who are the best known of the new groups in the US. Tower Recordings are also fairly well-known: Furniture Music for Evening Shuttles. Another group worth checking out is SHOTM (Sunburned Hand of the Man); Amazon only has one CD of theirs: Rare Wood. On the more mellow/acoustic side is Six Organs of Admittance: Dark Noontide. Also more acoustic, and probably a bit more in the realm of drone-rock are Pelt: Empty Bell Ringing in the Sky.
Leaving America, Japan has carried on the tradition of acid folk, most notably with the music of Ghost: Second Time Around. Another collective, the Acid Mothers Temple, is often very loud, but they do have a more mellow/minimalist/folk side: Mantra of Love. Finally, there is some great weird stuff coming out of England from the Vibracathedral Orchestra, The Queen of GuessDabbling With Gravity and Who You Are and Sunroof, Sound Star Sound.