Edie Brickell and New Bohemians’ song “What I Am” should get some love here.
I was going to link to their SNL performance… but this other one has cowbell.
Wow, three pages long and no one has mentioned *We’re Only in It For the Money *by the Mothers of Invention.
Yea, I know it’s a parody. But the music is hippie and great. 
Zappa’s Lumpy Gravy also has some great “hippie” tunes.
Other than that, the only thing I’ve got is Strawberry Alarm Clock, which others have already mentioned.
I came in specifically to mention this one.
Gary “Wombat” Robson, are you making a mix-tape? If so, can I have a copy? 
Hole in my Shoe by Traffic
Flowers in the Rain by The Move
Virtually anything on Seargent Pepper or Revolver.
A few rather obscure ones that I particularly love…
Essence of Nowby It’s A Beautiful Day.
"Maker"by the Hollies (just before Graham Nash left)
“Orange and Red Beams” by Eric Burdon & the Animals
“Putting Out the Vibration and Hoping It Comes Home” by Richie Havens
“Pictures and Designs” by the Seeds
“Birds Can’t Row Boats” by Johnny Winter
“Everybody’s Here” by Lost and Found
“Pindar’s Revenge” by Ed Sanders (watch out, this one is really out there)
“Kites” by Simon Dupree and the Big Sound
And here are some from more well-known artists:
Beatles – “Rain”, “It’s All Too Much”
Stones – “In Another Land”, “Child of the Moon”
Pink Floyd – “Flaming”, “Lucifer Sam”
Donovan – “Three King Fishers”, “The Fat Angel”
Moody Blues – “Legend of a Mind”, “Departure/Ride My See-Saw”
Cream – “World of Pain”, “Dance the Night Away”
This is helping me to assemble my playlist from many songs I already have, but there are dozens of songs mentioned that I don’t own and dozens more that I’ve never even heard of. I have quite a task ahead of me!
I’ll be previewing all of these and probably spending a bunch of money buying some of them.
Reaching a little deeper: while not as well known as Baez and Dylan, these folkies were very popular in hippie circles.
Buffy Sainte-Marie (who actually wrote “Universal Soldier”): Codine was one of my faves, along with Cripple Creek and Broke Down Baby. I saw her in concert in NYC in about 1968: front row seats!
Ian & Sylvia: Four Strong Winds and other tunes from the wilds of Canada.
Boobs a Lot - Fugs or Holy Modal Rounders
I Feel Like I’m Fixin’ to Die Rag - Country Joe and the Fish
If we’re talking Eric Burdon I’d have to throw San Francisco Nights and Sky Pilot in there too.
Humble Pie: 30 Days in the Hole.
I really want to put some Deep Purple and Tommy Bolin in there, like maybe off Come Taste the Band but I wonder if that’s not too post-hippy, like Post Toastee.
Thbbbt!
Nobody else remembers The Hombres? Let it all hang out, man.
Hm. Can’t find a copy of Paul Butterfield’s Love March, but it was pretty awesome too.
Problem with that song is the refrain about “warm San Francisco nights.” As a former Haight-Ashbury hippie, I can tell you, there ain’t no such, uh, animal!
Free to Be You & Me - the New Seekers
Yes, yes & yes, but you have to include Quicksilver’s “The Fool” (if you can listen to the ending instrumental choruses without holding your breath, you got me beat). John Cippolina & Gary Duncan - best hippy Guitar duo ever.
Don’t miss Buffy’s “God is Alive, Magic is Afoot”. Totally trippy.
Holy Modal Rounders, “If You Wanna Be a Bird”
Mungojerry, “In the Summertime”
Insect Trust, “The Eyes of a New York Woman”
No, no, no! The original song (“I’d like to teach the world to sing”) is already on my list, but we’re talking about hippie music. You can’t use the commercialized version with the words changed to promote a product. Absolutely not.
Buffy’s music–Leonard Cohen’s words.
Zager and Evans - “In the Year 2525”
T. Rex - “Get It On”