We report the structural and magnetic properties of rare-earth substituted spin-chain compounds Ca_2.75R_0.25Co₂O₆ (R = Dy and Lu). The Rietveld refinement of neutron and x-ray powder diffraction patterns confirms the single-phase formation of both compounds in the rhombohedral structure (space group R¯3c). The derived values (from the analysis of the neutron diffraction patterns at 50 K) of the bond-valence sum indicate a reduction in the oxidation state of the cobalt ions at the trigonal prism (TP) site (6a) with R substitution, which is further supported by low temperature neutron diffraction [where a zero value of the ordered moment at the 6b site and a reduction in the values of the maximum ordered moment at the TP site have been observed] and dc magnetization studies. In the neutron diffraction patterns, additional Bragg peaks appear for both compounds below N´eel temperature (T_N) of ˜16 K, indicating the onset of an antiferromagnetic ordering of cobalt spin chains on the triangular lattice. The magnetic structure corresponds to a spin density wave (SDW) structure [with a propagation vector k = {0,0,1.02}], having c axis as a direction of both moment and modulation. For both compounds, the refined values of the ordered moment at the 18e, 6b, and 6a sites are ˜0.03(2),0.02(2), and 4.2 (2) μB, respectively. Unlike the parent compound Ca₃Co₂O₆, no temperature dependence as well as no time dependence in the intensity of the strongest antiferromagnetic reflection (10 τ ), corresponding to the propagation vector k = {0, 0, 1.02}, has been observed down to 1.5 K confirming that the SDW structure is stabilized by the substitution with rare-earth ions. The stabilization of the SDW structure and the observed decrease in the values of T_N could be due to a decrease in the value of positive FM intrachain exchange interaction J with the rare-earth substitution in a system with competing intrachain and interchain exchange interactions.