Spoken Word

Speech used as song. During the 1960s and 1970s in particular, this technique was often used in film musicals to allow popular actors to star in them despite a total or near-total lack of singing ability. It is also used in theatre when production schedules simply do not allow time for proper rehearsal of choral numbers. It is also a style of performance grown out of Jazz and Soul which eventually influenced the evolution of Rap.

Examples of Spoken Word include:

Rex Harrison in My Fair Lady and Doctor Dolittle. He dubbed the technique "Sing-Speak".
Gregorian chant (in some cases) ^[context?]
Lorne Greene in Bonanza performs the theme's lyrics this way. He did the same on his much more famous recording, "Ringo".
John DiMaggio, voicing Gorilla Grodd in the Batman: The Brave And The Bold episode "Mayhem of the Music Meister!", uses this technique.
Just about everyone in the film version of Man of La Mancha, which is fortunate since none of the stars were even tolerable singers.
Johnny Cash
Robert Preston as Professor Harold Hill in The Music Man.
C.W. McCall's recordings from the 1970s -- starting with his big hit "Convoy" and including just about everything else he ever did -- is basically him reciting a lyrical, rhyming story with the occasional interjection by background singers; they weren't so much songs as a kind of country-western proto-Rap.
Used on The Muppet Show by most guest stars who couldn't actually sing. (Some tried anyway.)
The musical Camelot does this quite a bit.
"All I Wanna Do Is Play Cards" by Corb Lund.
Gil Scott-Heron is famous for his Spoken Word performances, most notably "The Revolution Will Not Be Televised".
Later recordings by William Shatner have him doing the lyrics in Spoken Word while the backup singers... well, sing.