Thanks to @Andrew I dug out the following solution from A. Friedman "Partial Differential Equations of Parabolic Type": $$ u(\vec{x},\vec{\mu},t) = \frac{\sqrt{|\mathbf{D}|}}{(4\pi~ t)^{d/2}} \exp \large(\frac{-(\vec{x}-\vec{\mu})^{T} \mathbf{D}^{-1}(\vec{x}-\vec{\mu})}{4~|\mathbf{D}|~t}\large). $$ It seems the root of the determinant of the diffusion tensor goes on top. Take it with a grain of salt.

Thanks to @Andrew I dug out the following solution: $$ u(\vec{x},\vec{\mu},t) = \frac{1}{(4\pi~ t)^{d/2}~\sqrt{|\mathbf{D}|}} \exp \large(\frac{-(\vec{x}-\vec{\mu})^{T} \mathbf{D}^{-1}(\vec{x}-\vec{\mu})}{4~t}\large). $$ It seems the root of the determinant is not dependend on the problem dim. Take it with a grain of salt.