CW answer since this has only been posted in comments. After this question was posted in 2011, Terry Tao has (in 2013) posted a proof reducing to character theory only of abelian groups.

From Tao's post (May 24, 2013):

I didn’t succeed in obtaining a completely elementary proof, but I did find an argument which replaces character theory (which can be viewed as coming from the representation theory of the non-commutative group algebra ${{\bf C} G \equiv L^2(G)})$ with the Fourier analysis of class functions (i.e. the representation theory of the centre ${Z({\bf C} G) \equiv L^2(G)^G}$ of the group algebra), thus replacing non-commutative representation theory by commutative representation theory.

Copy of Geoff Robinson's comment (May 29, 2013):

The new proof of Terry Tao is an alternative proof. It reduces the problem in an ingenious way to one of character theory of commutative semisimple algebras, though it is close in spirit to the original character-theoretic proof. It had not been devised at the time when the accepted post below was written. It is probably still fair to say that there is no purely group-theoretic proof of Frobenius's theorem.