In David Lynch's groundbreaking TV series Twin Peaks, this "clue" (such as it was) was delivered to Special Agent Dale Cooper by the Giant (actor Carel Struycken), who visited him in a dream while he slept in his hotel room at the Great Northern. This scene appeared in the episode entitled "May The Giant Be With You" (how appropriate) which originally aired on September 30, 1990.
Many consider this to be among the most memorable scenes from the series, but this quote is sometimes mistaken as belonging to to a scene that was in another episode ("Zen, or the Skill to Catch a Killer", April 19, 1990). This scene, probably the most jarring one of the entire series, is the one containing the backwards-talking dancing midget, referred to in the series as The Man From Another Place and played by actor Michael J. Anderson.
The dream sequence from the end of this episode is far more amazing than the Owls scene in the Giant episode; in it, the dwarf reveals to Cooper: "That gum you like is going to come back in style." He introduces Laura Palmer to Cooper in the dream ("She's my cousin... but doesn't she look... almost exactly like Laura Palmer?"), and she whispers in Cooper's ear the name of her killer, which Cooper can not seem to remember upon waking. This scene was very surreal and made a lot of viewers freak out because it was quite unlike anything that had ever been seen in a prime-time TV series before.
Thanks to fnordian for a minor factual correction.
Complete transcripts of the two episodes can be found here: