They’re Coming to Take Me Away, Ha-Ha! (Napoleon XIV). It’ll never make it on the air—'way too un-PC—but I do remember it fondly.
Or they could play side B, which is the same song backwards.
I’ve gotten so that I rarely listen to regular radio these days: it’s pretty much XM radio only for me. And because of that, a lot of the songs I see mentioned here I’ve heard relatively recently.
Heck, last week I heard “Transfusion” by Norvus Nevous (on the 50s channel), which was the first time I’ve heard it outside of a Dr. Demento show!
I’d add to the playlist songs by groups that had at most a handful of hits (or perhaps they were one hit wonders), but weren’t the song(s) that made the hit list. Similar to NDP’s requests for every song from Otis Redding but ‘Dock of the Bay’, but expanded.
I would also add regional hits: songs and/or groups that were popular in one area, but nearly unheard of outside of that region. Garage bands, etc. Some of the Carolina Beach Music would fall under this heading.
<< Don’t worry–be silly! >>
Oh my God, Ellen, that’s some great schlock! About the only thing I can add to it is “How do You Do”, by Mouth and MacNeal. ("and then we can na-na-na-na . . . )
The mention of "Bang A Gong"brings to mind:
Bang A Gong – Power Station
and while we’re at it
Some Like It Hot – Power Station
and also also while we’re at it
Sneakin’ Sally Thru The Alley – Robert Palmer
but absolutely not “Addicted to Love” or “Doctor, Doctor” by Palmer. Overplayed. Which is a same, because Dr, Dr is really good.
That’s because none of Dick Dale’s singles ever made the national Top 40. He was a regional/cult favorite whose style was highly influential but who never got the widespread public attention that, for example, the Ventures did. The only reason people know about him now is because Tarantino used “Miserlou” for Pulp Fiction’s opening credits. Oddly enough, this led to “Miserlou” probably getting more airplay in 1994 than it did when it first came out in the early 60’s.
The second that blared out of the screen, I turned to my date and said “We are in good hands.” 
Our oldies channel needs a surf/sand/cars/cuties show at least weekly. 4 hours of nothing but surf music, drag racing songs, and odes to someone’s girlfriend. All produced by groups you never heard of in someone’s garage for $40 that managed to sell enough copies to friends to chart for a week at #97 in 1964.
Alright, how about original versions of great songs that are never heard on the air. Such as:
“Blue Suede Shoes” - Carl Perkins
“Shake Rattle and Roll” - Big Joe Turner
“Not Fade Away” - Buddy Holly
And I propose a trade - in return for permanently retiring everything Paul McCartney and Wings ever recorded, we get to hear “I’m Down” by the Beatles.
Napster, iTunes et al were *invented * for this purpose. I have been soooo happy, remembering a song I used to love, and going out and gettin’ it.
Good call on Hey Bulldog. One of the first ones on my list.
**Have You Seen Your Mother, Baby, Standing in the Shadow ** - Rolling Stones. Stones are a staple of oldies stations, but this song *never * gets played.
Does “oldies” no longer refer to late-50s and early-60s music? I thought a lot of the suggestions here would be more appropriate for “classic rock” and “album rock” stations.
My contributions:
Pipe Dream, by Chad & Jeremy
Soul Pad, by the Coasters
When Something is Wrong with My Baby, by Sam & Dave
Let’s Go Get Stoned, by Ray Charles
Your Auntie Grizelda, by the Monkees
Stranded in the Jungle, by the Cadets
Marmalade - Reflections of My Life
New Colony Six - I Will Always Think About You
The Left Banke - Pretty Ballerina
The Loving Spoonful - Darling Be Home Soon
Matthew’s Southern Comfort - Woodstock
Poppy Family - Which Way You’re Going, Billy?
Friend and Lover - I Think It’s So Groovy Now(That People Are Finally Getting Together)
Love - Orange Skies
I’d also want the station to have dedicated days. XM had an “All Monkees” day on one of their channels last week. Let’s hear nothing but Yardbirds for 6 hours every now and then.
Black Light Trap by Shreikback
This is so much fun. I’m totally down with the Which Way You Goin’ Billy? suggestion. I haven’t thought of that song in years.
Cecelia, Simon and Garfunkel (I do hear this occasionally, but I could listen to this and The Boxer over and over and over)
I’ll Have To Say I Love You In A Song, Jim Croce
My Baby Loves Lovin’, White Plains
Down In The Boondocks, Billy Joe Royal
Harper Valley P.T.A., Jeannie C. Riley
Look What They’ve Done To My Song Ma, New Seekers
In The Ghetto, Elvis Presley (The scholckiest of 'em all)
The Night Chicago Died, Paper Lace (Well, maybe this is)
Rock Me Gently, Andy Kim
Working In The Coal Mine, Lee Dorsey
Worst That Could Happen, Brooklyn Bridge
Billy Don’t Be A Hero, Bo Donaldson and The Heywoods
Rock The Boat, Hues Corporation
Wildwood Weed, Jim Stafford (This or Spiders ‘n Snakes? So hard to choose)
These Boots Are Made For Walkin’, Nancy Sinatra
Bang Bang, Cher (When was the last time YOU thought of this song? :D)
I’ll restrict mine to Billboard Top 20 hits:
Keem-O-Sabe- Electric Indian
When I Die- Motherlode
Oh Babe What Would You Say- Hurricane Smith
Don’t Say You Don’t Remember- Beverly Bremers
Driver’s Seat- Sniff ‘N The Tears
I Like Dreamin’- Kenny Nolan
Do it Again a Little Bit Slower- Jon & Robin & the In Crowd
Hold On- Ian Gomm
Eres Tu- Mocedades
Toast & Marmalade for Tea- Tin Tin
Good Time Charlie’s Got the Blues- Danny O’ Keefe
Last Song- Edward Bear
One Fine Morning- Lighthouse (ok, Top 30)
Yellow River- Christie (ditto)
Stay Awhile- The Bells
Joy- Apollo 100
Popcorn- Hot Butter
Lead Me On- Maxine Nightingale
one more- Ariel, Dean Friedman (not Billy Joel) 
And I really hope that the reason Sonja Kristina & Curved Air’s “Back Street Luv”
didn’t chart in the US was because it wasn’t issued as a single, and not because of listner apathy.
Great call(s) on “Some Velvet Morning”, quite possibly the weirdest song ever to make the Top 40.
I’d love to hear two of my favorite '50s rockers, “Rip It Up” and “Ready Teddy,” on the radio. Elvis Presley and Little Richard, among others, did great versions.
This is one of my favorite albums.
And I’ll add anything by Robin Trower to this list. I love Bridge of Sighs.
Actual title: “Reach Out of the Darkness”
I’ll throw in:
Crow, “Evil Woman” (not to be confused with the 1,247 other songs out there called “Evil Woman”)
Bubble Puppy, “Hot Smoke and Sassafras”
Southwest F.O.B., “Smell of Incense”
Colours, “Love Heals”
Turtles, “Sound Asleep”
Iron Butterfly, “Soul Experience”
NDP, props for spelling Procol Harum correctly, and for the excellent Ray Charles selection. Jean Poutine, “Back Street Luv” was released as a single in the USA.
Much thanks to Evil Captor for this great thread. I had thoughts of starting a similar thread, but, in responding to this one, I’m reminded of another thread idea I had - picking a favorite year for hit singles. My favorite year was 1972, and for this thread I went to “The Book of Rock Lists” to refresh my memory and find these great songs that no longer get sufficient airplay:
“Papa was a Rolling Stone”-THE TEMPTATIONS
“Freddie’s Dead”-CURTIS MAYFIELD
“If You Don’t Know Me By Now”-HAROLD MELVIN &THE BLUE NOTES
“Mother & Child Reunion”-PAUL SIMON
“(If Loving You Is Wrong)I Don’t Want to Be Right”-LUTHER INGRAM
“Betcha By Golly, Wow”-THE STYLISTICS
"Superfly"CURTIS MAYFIELD
“I Saw the Light”-TODD RUNDGREN
“Back Off Boogaloo”-RINGO STARR