Considering all the answers so far, I thought I might as well add one with a more topological flavor to it

 1. The existence of an L-space was known to be consistent for years (See The Handbook of Set-Theoretic Topology, Chapter 7, pg. 295). It was only recently that a ZFC construction was given [here][1]

 2. Some of the existence proofs for certain types of embeddings, and automorphisms between Boolean algebras have this flair to them, (See "The fourth head of $\beta\mathbb{N}$" by Ilijas Farah, in Open Problems in Topology II. pg 139.)

 3. Certain types of gnarly questions about coverings of $\mathbb{R}$ involving the forward and inverse images of $\aleph_1$ many continuous functions, have had some success with this [see this answer][2]


  [1]: http://www.math.cornell.edu/~justin/Ftp/Lspace.pdf
  [2]: http://mathoverflow.net/questions/53711/cn-and-forcing-reading-a-recent-paper-by-kunen/53728#53728