Under ZFC, the negation of CH is equivalent to [Freiling's "Axiom of Symmetry"][1].  Wikipedia says this equivalence was proven by Sierpinski.  

Freiling's axiom is fairly intuitive, in that it basically says that (countably) discrete invariants on the reals disagree at points, for "probabilistic" reasons.  I think it makes a pretty cool argument against CH :)


  [1]: http://en.wikipedia.org/wiki/Freiling's_axiom_of_symmetry