Our new book
Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids, EMS Tracts in Mathematics vol 15 http://pages.bangor.ac.uk/~mas010/nonab-a-t.html
uses mainly cubical, rather than simplicial, sets. The reasons are explained in the Introduction. In strict cubical higher categories we can easily express
algebraic inverse to subdivision,
a simple intuition which I have found difficult to express in simplicial terms. Thus cubes are useful for local-to-global problems. This intuition is crucial for our Higher Homotopy Seifert-van Kampen Theorem, which enables new calculations of some homotopy types, and suggests a new foundation for algebraic topology at the border between homotopy and homology.
Also cubes have a nice tensor product and this is crucial in the book for obtaining some homotopy classification results.
I have found that with cubes I have been able to conjecture and in the end prove theorems which have enabled new nonabelian calculations in homotopy theory, e.g. of second relative homotopy groups. So I have been happy to use cubes until someone comes up with something better. ($n$-simplicial methods, in conjunction with cubical ideas, turned out, however, to be necessary for proofs in the work with J.-L. Loday.)
See also some beamer presentations available on my preprint page.