[Some of the links in the comments are dead.]
Fundamental solutions of the wave equations are truly singular, and before distributions it was almost impossible to describe. If you don't care about the constant in front, it can be described by repeated (distributional) derivatives of (regular) functions.
For n = odd, let m = (n-1)/2 and
$ E = \square^m H(t-|x|) $
but for n = even, with m = (n-2)/2
$ E = \square^m \frac{H(t-|x|)}{\sqrt{t^2-|x|^2}} $
Some remarks: In the first case, $E$ is supported on the surface of the (future) cone, and you can think of it as some higher derivative of the "$\delta$-function" along the surface. For the second case, the distribution is regular in the interior of the cone (and is $(t^2-x^2)^{-\frac{1}{2}-m}$ there); but this blows up along the boundary. What the distribution above tells you is how to "remove" the infinity. This is precisely what inspired Hadamard's theory of partie finie, which in turn was one of the motivations for Schwartz's distribution theory.