There are surprisingly few references to "Sparse Diagonal Storage" format on Google. Here is a PDF of some slides I found that provide a brief overview of Compressed Row, Jagged Diagonal, and Sparse Diagonal formats. Most of the other links are pointers to a single paper, Use of sparse diagonal storage in finite element analysis by
K.K. Yalamanchili and S.C. Anand.
So this isn't good news, really: It suggests that Sparse Diagonal Storage has not been adopted very widely. I'm going to hazard a guess: the format is probably more useful for iterative methods than for direct (factorization) methods. Implementing sparse matrix-vector multiplication is relatively straightforward for any of these formats, so for iterative methods you have the freedom to store as you wish (at least until you need an incomplete factorization for a preconditioner!)
If your matrix has a relatively dense diagonal band, then you may want to consider one of LAPACK's banded solvers. These store the diagonals along the rows of a matrix with B rows, where B is the bandwidth of the matrix. (EDIT: or (B+1)/2 rows if the matrix is symmetric/Hermitian). But if you truly require a sparse direct solver, then it looks like you will either have to implement your own or convert to a more common storage format. (If you decide to implement your own, perhaps you will quickly learn why SDS is not suitable :P)
EDIT: The abstract for the Yalamanchili-Anand paper suggests that they are focused on vector-matrix multiplication; i.e., iterative methods.