Summary: For any linear inequality in three variables \(\mathcal{L}\), we determine (if it exists) the smallest integer \(R(\mathcal{L},\mathbb{Z}/3\mathbb{Z})\) such that: for every mapping \(\chi:[1,n]\rightarrow\{0,1,2\}\), with \(n\geq R(\mathcal{L},\mathbb{Z}/3\mathbb{Z})\), there is a solution \((x_1,x_2,x_3)\in[1,n]^3\) of \(\mathcal{L}\) with \(\chi(x_1)+\chi(x_2)+\chi(x_3)\equiv 0\pmod{3}\). Moreover, we prove that \(R(\mathcal{L},\mathbb{Z}/3\mathbb{Z})=R(\mathcal{L},2)\), where \(R(\mathcal{L},2)\) denotes the classical 2-color Rado number, that is, the smallest integer (provided it exists) such that for every 2-coloring of \([1,n]\), with \(n\geq R(\mathcal{L},2)\), there is a monochromatic solution of \(\mathcal{L}\). Thus, we get an Erdős-Ginzburg-Ziv type generalization for all linear Diophantine inequalities in three variables having a solution in the positive integers. We also show a number of families of linear Diophantine equations in three variables \(\mathcal{L}\) which do not admit such Erdős-Ginzburg-Ziv type generalization, named \(R(\mathcal{L},\mathbb{Z}/3\mathbb{Z})\neq R(\mathcal{L},2)\). At the end of this paper some questions are proposed.