Contraction, American
accent, or mispronunciation for the phrase "don't know".
E.g.,
I dunno, but it's WAY famous dude!
=
I don't know, but it's WAY famous dude!
The above phrase was heard uttered at the Louvre in 1988 by two California surfer dudes admiring the Victory of Samothrace.
Added March 19, 2002:
According to the Oxford Dictionary of Modern Slang (John Ayto and John Simpson, eds., NY:Oxford Univ. Press, 1992, p. 61):
"Representing a casual pronunciation of (I) Don't know."
Initial use was in 1842 as noted in the main entry for dunno in the Oxford English Dictionary (2nd Ed.).