Cases of nearly linear time solvable linear systems
Linear systems with circulant matrices can be solved in $O(n \log n)$ using the fast Fourier transform. A circulant matrix has entries $a_{i,j} = a_{1,i+j-1 \bmod n}$ so it is completely specified by the first column. Circulant matrices are diagonalized by the fast Fourier transform (log-linear time) and linear systems with diagonal matrices can be solved in linear time. See for instance the Wikipedia article http://en.wikipedia.org/wiki/Circulant_matrix for more details.

Apologies if this is too trivial to be mentioned here.