Let \(S\) be a Damek-Ricci space. The first main result of the paper states that \(S,\) endowed with the right Haar measure \(\rho\) and the left invariant metric, is a Calderón-Zygmund space. The second result is a Hörmander type theorem for spectral multipliers associated to the Laplacian \(\Delta.\) Let \(\psi\in C_c^\infty(\mathbb{R}^+)\) be supported in \([1\slash4,4]\) and, for every \(\lambda\in\mathbb{R}^+,\) \(\sum_{j\in\mathbb{Z}}\psi(2^{-j}\lambda)=1.\) Let \(m\) be a bounded measurable function on \(\mathbb{R}^+.\) Define \(\| m\| _{0,s}=\sup_{t3\slash 2,\) \(s_\infty>\max\{3\slash 2,n\slash 2\}\) and \(\| m\| _{0,s_0}<\infty,\) \(\| m\| _{\infty,s_\infty}<\infty.\) Then \(m(\Delta)\) is bounded from \(L^1(\rho)\) to \(L^{1,\infty}(\rho)\) and on \(L^p(\rho)\) for all \(p\in(1,\infty).\) The proof is based on an \(L^1\)-estimate of the gradient of the heat kernel associated to \(\Delta.\)