There's also this stunning performance of one of John Barry's last songs: Tick the Days - by John Barry - sung by David Arnold
It doesn't sound like a Bond theme song, but it sounds very much like the love ballads Barry wrote for some Bond films, like "We Have All The Time In The World" and "If There Was A Man".
"Bits & Pieces" from THE STUNT MAN comes to mind. Music by Dominic Frontiere, lyrics by Norman Gimbel, sung by Dusty Springfield.
Lots of great finds here, but this one is my favorite. Loved the movie since it was finally released in 1980, had the record at the time and the Intrada as soon as they released, but the link to a Bond song never occurred until now.
"With Every Breath I Take" from the musical "City of Angels" (Music by Cy Coleman, Lyrics by David Zippel) always struck me as the best Bond song not written for a James Bond movie. The arrangement is in more of a 40s style (which makes sense as that's when the show is set), but you can easily mentally Barryize it. The chord progression is pretty Bondian (and you can hum some of those Barry rat-tat-tat-tat-tat brass interjections during the line "I'm embracing you"), and the lyrics have that sort of knowing torchiness that characterizes a lot of my favorite Bond tunes.
A lot of the pseudo-Bond films had pseudo-Bond songs. One of the better ones was WHERE THE BULLETS FLY (1966), which was scored by Kenny Graham. Its Bondish title song was sung by Susan Maughan.
Thanks for all the replies, everyone. I should have remembered "Bits and Pieces" from The Stunt Man! Dusty Springfield has that "thing" about her singing just like the "thing" Shirley Bassey has.
I want to steer away from spoofs and parodies--though they are fun songs in their own right--but I am enjoying the song examples that were based on the scores but unused, as well.
I'm off to listen to the rest of everyone's suggested songs right now. Just one thing: if you embed a video, please post the title in case said video suddenly vanishes; the internet being so damned temporary and transient and all.
Amy Winehouse would have been so perfect. Such a shame that she never got to record her song. I just checked the Jon Burlingame book, and he points out that Arnold originally wrote the 6 note Quantum of Solace theme for the abandoned Winehouse song, which she never finished writing and never showed up to record. Arnold later used it in the film's score, and then built the song "No Good About Goodbye" around it.
Her sound was undeniably Bond. Here, someone took her hit "Back to Black" and synched it up with the Casino Royale credits.
In listening to these songs, I try and picture the Maurice Binder-style opening credits with the song playing. If it succeeds on that level, it's essentially "in."
I'm making my way through all the songs...
The Cheap Trick song, while definitely "George Martin sounding", doesn't really work for me, though it *is* close. Maybe if the intro were shortened and then straight into the song. Yes, I am quite the music producer.
The Amy Winehouse song is really good and yes, a shame that it never happened.
The Last Shadow Puppets one is brilliant! Like some lost, alternate reality 1966-67 that should have happened. The vocalist sort of reminds me of Gerry Marsden.
I like how the selections can go with specific eras of Bond. It doesn't all have to be Bassey. I can easily picture these tunes going with some never-made Timothy Dalton or Roger Moore (gasp) entry.
Just one thing: if you embed a video, please post the title in case said video suddenly vanishes; the internet being so damned temporary and transient and all. 
