An \(L(2,1)\)-labeling of a graph \(G\) is a function \(f\) from the vertex set \(V(G)\) to the set of all nonnegative integers such that \(|f(x) – f(y)| \geq 2\) if \(d(x,y) = 1\) and \(|f(x) – f(y)| \geq 1\) if \(d(x,y) = 2\), where \(d(x,y)\) denotes the distance between vertices \(x\) and \(y\) in \(G\). The \(L(2,1)\)-labeling number \(\lambda(G)\) of \(G\) is the smallest number \(k\) such that \(G\) has an \(L(2,1)\)-labeling with \(\max\{f(v) : v \in V(G)\} = k\). We consider Cartesian sums of graphs and derive, both, lower and upper bounds for the \(L(2,1)\)-labeling number of this class of graphs; we use two approaches to derive the upper bounds for the \(L(2,1)\)-labeling number and both approaches improve previously known upper bounds. We also present several approximation algorithms for computing \(L(2,1)\)-labelings for Cartesian sum graphs.