Dear Donu, here are Grothendieck's own words:

"Contrary to what occurs in ordinary
topology, one finds oneself confronting
a disconcerting abundance of different
cohomological theories. One has the
distinct impression (but in a sense that
remains vague) that each of these theories
“amount to the same thing”, that
they “give the same results”. In order to
express this intuition, of the kinship of
these different cohomological theories,
I formulated the notion of “motive” associated
to an algebraic variety. By this
term, I want to suggest that it is the
“common motive” (or “common reason”)
behind this multitude of cohomological
invariants attached to an
algebraic variety, or indeed, behind all
cohomological invariants that are a
priori possible"

They can be found in his autobiographical "Récoltes et Semailles", where there is also an allusion to a musical meaning of "motif". 

The translation is Barry Mazur's in his article "What is... a Motive?" which is, needless to say, a fascinating short survey (plus bibliography) .Here is the reference:

http://www.ams.org/notices/200410/what-is.pdf