Though I heartily agree with Victor Protsak's comment, I will add some references.  These might be useful for you, at least if you haven't seen them before.  We must add a restriction, however, by assuming that $I$ is an ideal of finite co-length.

Then Corollary 12.5 of Eisenbud's Commutative Algebra uses the theory of Hilbert-Samuel polynomials to prove that $\text{dim}(R)=\text{dim}\text{ gr}_I(R)$.  

Alternately, you might also be interested in Corollary 10.12 of the same book.  This second corollary assumes that $I=\mathfrak m$, but the proof makes use of "Going down for flat extensions", which has a somewhat different flavor than the Rees ring approach.