Well, what immediately came to mind for me was the song, Summertime. Janis Joplin did a cover of it, but I think she was influenced by Big Mama Thorton, and other artist have covered the song as well (both instrumentally and vocally). I might also suggest a jazz CD that has the same era feel to it, and a very good CD to buy, and that is: Charlie Haden & Quartet West's Haunted Heart. It's done in the style of the 40's era, and has vocals on it by Jo Stafford, Jeri Southern, and Billie Holiday on it as well. They made a sequel CD and it is equally worth picking up titled, Always Say Goodbye.
I might also mention a program I listen to on National Public Radio called Swinging Down the Lane. It's great and has all sort of good music on it and an informed emcee. You can see if it plays in your area. Unforunately the program does not have an archive to listen on line, or at least it didn't the last time I looked. Here's a link: http://www.bigband-era.com/
I found a couple of other musicians/bands that have covered Summertime. Stan Getz does a version, and has played many other great songs: The Girl From Ipanema (might be good for summer), and Ella Fitzgerald & Louis Armstrong do a vocal version of Summertime as well.