from this article: http://web.stanford.edu/class/math220a/handouts/waveequation3.pdf,
He computed odd and greater than 1 dimension using spherical mean, and for even dimension used method of descending. because of that, the Huygens principle holds for odd and greater than 1 dimensions, but doesn't for even ones.
why only odd ones he used the spherical mean?
The reason is that there is not a transformation such as rM that allows you to reduce the equation of Euler Poisson Darboux to a wave equation in one dimensión. You can make the computations in d=2, the transformation rM does not work. And in fact the proof does depend on the dimension. If you keep Reading Evan's book you will find the correct transformations for general odd dimensions because rM doesn't do the trick.