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.

Examples:

Rex Harrison in My Fair Lady and Doctor Dolittle. He dubbed the technique "Sing-Speak".
Gregorian chant (in some cases)
Lorne Greene in Bonanza performs the theme's lyrics this way.
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
Used on the Muppet Show by any guest star who couldn't actually sing.
The musical Camelot does this quite a bit.
"All I Wanna Do Is Play Cards" by Corb Lund.