No Regrets was written by folk singer Tom Rush and was the closing track on his 1968 album The Circle Game
It was more famous for folk of a certain age for a cover in 1976 by The Walker Brothers
... or for us slightly younger folk, for a cover by Midge Ure in 1982
Luna's cover appeared in 1995 as one of the tracks on the Hedgehog/23 Minutes in Brussels single, and later on the No6 Records EP (and more recently on the Lunafied download only compilation).
Audio: Luna - No Regrets
And as a bonus here's Lee Hazlewood's version from The Cowboy and The Lady from 1969