Method to compute fundamental solutions which are distributions The Malgrange-Ehrenpreis theorem tells us that there is a fundamental solution for any linear differential operator of constants coefficients. The original proof was not constructive (it was based on the Hahn-Banach theorem). Lars Hörmander first gave a method (Hörmander's staircase) to explicitely compute fundamental solutions. Nevertheless, the solutions his method provides are usually "too bad", for they are not even tempered distributions.
Therefore, a natural question arises: Is there a general method  to explicitely construct fundamental solutions which are actually tempered distributions?
 A: The fact that there always exists a temperate fundamental solution was proved by Hormander here in 1958. I don't know if this answers some part of your question. Concerning regularity, well, this is an interesting question (most of the classical theory of PDEs is concerned with it). 
Let me also mention a nice paper by 
Rosay who reproves Malgrange-Ehrenpreis in the $L^2$ setting with an elementary proof, but still non constructive
A: If you are interested in explicit fundamental solutions for pde's, you should take a look at the supremely interesting work of the Innsbruck group (P. Wagner and N. Ortner).  Detailed references can be found via Google. For starters there is their "A survey on explicit representation formulae for fundamental solutions of linear partial differential operators" in Acta Applicandae Math. 47 (1997) 101-124.
A: Another approach uses Bernstein-Sato polynomials. If $P(s,\partial)$ is a differential operator with polynomial coefficients and $f(x)\in\mathbb{C}[x_1,\ldots,x_n]$ then the Bernstein-Sato polynomial for $P$ and $f$ satisfies
$$P(s,\partial)f(x)^{s+1} = b(s) f(x)^s$$
If $f$ has only nonnegative values on $\mathbb{R}^n$, this can be used to define an meromorphic continuation of the distribution-valued, holomorphic function $\{Re(s)>0\}\to\mathcal{S}'(\mathbb{R}^n), s\mapsto f(x)^s$ and in particular a distributional inverse for $f(x)$. (If $f$ doesn't have nonnegative values on $\mathbb{R}^n$, $f(x)\overline{f(x)}$ does and that can be used instead)
Distributionally inverting polynomials is by Fourier transform equivalent to finding Green functions for constant-coefficient PDEs
