Ok, this I know.  Viennot basically invents lots of great stuff but rarely publishes his work.  About "heaps of pieces" - this is a pretty little theory with very few original consequences.  It is really equivalent to Cartier-Foata partially commutative monoid (available <a href="http://www.mat.univie.ac.at/~slc/books/cartfoa.html">here</a>).  See Christian Krattenthaler's <a href="http://www.mat.univie.ac.at/~kratt/artikel/heaps.html">article</a> for the connection and details.  See there also some references to other recent papers.  

Now, for many of Viennot's unpublished results, see his "Orthogonal polynomials..." book, which was unavailable for years, but is now on his <a href="http://www.xavierviennot.org/xavier/Bienvenue.html">web page</a>.  See there also several of his video lectures (mostly in French), where he outlined some interesting bijections based on the heaps (some related to various lattice animals were new to me, even if he may have come up with them some years ago - take a look).  The reciprocal of R-R identities is an elegant single observation which he published separately: [MR0989236](http://www.ams.org/mathscinet-getitem?mr=989236).  It really does not reprove the R-R identities, just gives a new combinatorial interpretation for one side using heaps of dimers (various related results were obtained by Andrews-Baxter a bit earlier).  

About some recent applications outside of enumerative combinatorics.  Philippe Marchal describes <a href="http://algo.inria.fr/seminars/sem00-01/marchal.html">here</a> that heaps of pieces easily imply <a href="http://dbwilson.com/ja/tau.ps">David Wilson's theorem</a> on <a href="http://en.wikipedia.org/wiki/Loop-erased_random_walk">Loop-erased random walks</a> giving random spanning trees.  Ellenberg and Tymoczko give a <a href="http://arxiv.org/abs/math.GR/0510506">beautiful application</a> to the diameter bound of certain Cayley graphs.  Finally, in <a href="http://arxiv.org/abs/math/0607737">my paper</a> with Matjaz Konvalinka, we use a heaps-of-pieces style bijection to give the "book proof" of the (non- and q-commutative) <a href="http://en.wikipedia.org/wiki/MacMahon_Master_theorem">MacMahon Master theorem</a>.  

On your followup question regarding Kasteleyn's theorem and the Aztec diamond theorem - no, these are results of different kind, heaps don't really apply, at least as far as I know.