July

UK psychedelia from 1968, their self-titled LP. Where the rest of the album has a more predictable, albeit good, late 60's psych-pop sound, "The Way" sounds like a funky dub remix of a 60's song that could have been recorded yesterday, except that it is a 60's song, and it's actually not dubbed.
"The Way":

"I See":

Dream

Swedish psychedelic group Dream with their album Get Dreamy (1967).
"Do You Dream?"

Gandalf

american psych band, Gandalf with "Tiffany Rings" from their 1969 self-titled album:

Strawberry Alarm Clock

Though mostly considered to be a one-hit-wonder, Strawberry Alarm Clock released three albums from 1967-1968, during the height of the psychedelic movement in LA. They greatly influenced the early beginnings of LA bubblegum pop, which would grow to outlast and out-sell west coast psychedelia. The band broke apart in late '68, but released a 4th album, Good Morning Starshine, in '69 with an almost completely new line-up and a distinctly less psychedelic sound.
"Nightmare of Percussion" from their second album, Wake Up... It's Tomorrow (1968):

"Heated Love" from their 3rd album, The World in a Sea Shell (1968):

United States of America

Formed by electronic composer/musician Joseph Byrd at UCLA in 1967, United States of America artfully blended psychedelic sounds with chamber music, electronic experimentation, and avant-garde rock n roll. Their self titled LP (1968) was their only release.
The two songs I've selected showcase their soft rock meanderings mixed with mesmerizing tape and vocal effects and a calmer touch of psychedelia.
"Love Song for the Dead Che"

"Cloud Song"

The Beach Boys

Before Brian Wilson retreated from the public eye in the late 60's, he and the rest of the Boys had been busy in the studio recording sessions for their new album, SMiLE (or Dumb Angel), which was intended to follow up Pet Sounds (1966). Wilson was pouring himself into writing songs for the album which he called "a teenage symphony to God," but for whatever reason, before the sessions could be mastered and arranged for release, Brian Wilson scrapped the entire project and left the band, taking most of the songs with him. Rumors state that Capitol Records and some of the other Beach Boys disliked the new direction Brian was taking the band, that Brian was addicted to LSD, and even that Brian was crushed after being shown a demo of "A Day in the Life" from Sgt. Pepper's Lonely Hearts Club Band and couldn't imagine ever topping it. Many of the songs from the SMiLE sessions were reworked by Capitol and released on subsequent albums, but the original '66-'67 sessions with Brian Wilson will remain unreleased until this year.
"Heroes and Villains"

"Wonderful"

"Cabinessence"

"Surf's Up"

White Noise

In 1968, David Vorhaus formed White Noise with Delia Derbyshire and Brian Hodgson. The three were pioneers in early British electronic pop/psychedelic music, and produced their legendary album An Electric Storm later that year. After the record, Derbyshire and Hodgson went on to successful careers in composing and producing electronic music, and Doavid Vorhaus has continued on under the White Noise pseudonym, releasing 4 more albums into the 21st century.
"Firebird":

"Love Without Sound":

"My Game of Loving":

Further

Further was an LA group from the early 90's who's influence is muted at best (especially in light of the current music trends coming from the area) despite the fact that members of the group have deep rooted connections with modern genre benders and trend setters such as Beachwood Sparks and Ariel Pink's Haunted Graffiti. The following songs are from Further's 3rd and 4th album releases, Grimes Golden and Sometimes Chimes, both from 1994.
"California Bummer" from Grimes Golden:

"Quiet Riot Grrrl" from Grimes Golden:

"Surfing Pointers" from Sometimes Chimes:

The Beach Boys

...are most immediately known for their unmistakable harmonies, and perhaps especially in the case of the ground-breaking album, Pet Sounds (and Brian Wilson's withdrawl following its successful release). But one of the most ambitious and strikingly "different" songs on Pet Sounds contains none of their signature harmonies, or any voices at all for that matter...
"Let's Go Away For Awhile" from Pet Sounds (1966):

Mid Day Rain

1970 single from Mid Day Rain, their only two recordings I can find...
"Welcome to the Rain":

"Friday Mourning":

Brigitte Bardot Et Serge Gainsbourg

Brigitte Bardot, French singer/actress, teamed up with musician/actor/director Serge Gainsbourg to make several recordings in the late 60's. Gainsbourg's experimental, widely varied sound greatly influenced French popular and avant-garde music.
The duo originally recorded the "Bonnie and Clyde" single in 1967, and Gainsbourg released it in 1968:

Love

Arthur Lee and Love with "Old Man" from their classic 1967 album, Forever Changes:

Sun Ra

anyone who has an affinity to the absurd, psychedelic, or metaphysical and hasn't yet heard of Sun Ra ought to do some immediate research. This is from 1956:

Silver Apples

Silver Apples were a home-made synthesized rock band from the height of the flower power days. their third album, The Garden, was recorded between '68 and '69 but was unreleased until 1998.
"Again" from The Garden:

Cromagnon

from the album Orgasm (1969), considered by some to be a prophetic vision of black metal:
"Caledonia"

"First World of Bronze"

Judy Henske & Jerry Yester

Judy Henske had made a name for herself in Hollywood as a singer/songwriter and part-time actress (she was with Woody Allen for a while). She met Jerry Yester, a respected folk musician himself, in 1963 and they began making music together occasionally. Judy and Jerry were married at the time they recorded Farewell Aldebaran (1969) for Frank Zappa's Strait Records. The couple eventually split up and went their separate ways musically.
"Rapture" from Farewell Aldebaran

Lost Gringos

from Nippon Samba (1982)
Vida de Ines

Nippon Samba

Tambo Machay

Ohne Dich

Love

Arthur Lee is most known for his work with the band Love in the late 60s. it's seriously difficult to find vintage video of them online. just go listen to the music.

love arthur lee
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Frank Zappa & The Mothers of Invention

zappa and the mothers, live studio session for the BBC, 1968. ignore the repetition of the opening bit...: