Part of the difficulty is convincing yourself that you have made the correct geometrical dissection. In 2 and 3 dimensions, you can just look at the diagram and "know" you have it correct.
As an elementary problem of this sort, give a concrete geometrical proof that homotopy induces equality in singular homology. The basic difficulty [is to find a dissection of a prism into simplices]. But cranking out the explicit description, and proving that it works, in terms of all the basic definitions, is a wee bit of a mess.
I seem to recall a slick (but unmotivated) trick around this, called the method of cyclic models. Can one do something similar in the n-category case?