Four magnetic samples with different building blocks and dimensionalities, {[Cu₄(atr)₂(μ₃-OH)₂(sip)₂]·4H₂O}_n (1), {[Cu₄(atr)₂(H₂O)₈(μ-OH)₂(sip)₂]·1.3CH₃OH·0.7H₂O}_n (2), {[Cu₃(atr)₄(H₂O)₂(sip)₂]·4H₂O}_n (3) and {[Cu₃(atr)₄(H₂O)₂(μ₃-OH)(sip)(Hsip)]·2CH₃OH·2.75H₂O}_n (4) (atr = 4-amino-1,2,4-triazole and sip³⁻ = 5-sulfoisophthalate), were obtained and characterized structurally and magnetically. Complex 1 exhibits a three-dimensional (3D) robust framework with butterfly-like Cu^II₄ clusters periodically extended by tetratopic sip³⁻ connectors. Complex 2 possesses a 2D layer with alternating Cu^II₃ + Cu^II₁ chains crosslinked by pairs of ditopic sip³⁻ linkers. By contrast, the latter two entities feature 1D broad ribbons with linear (for 3) and triangular Cu^II₃ cores (for 4) propagated by bidirectional sip³⁻ connectors. Structural analysis reveals that the diverse building blocks and dimensionalities of 1–4 are significantly dominated by the tunable coordination of exocyclic amino- and/or sulfonate-group of the mixed ligands. Magnetically, antiferromagnetic interactions with variable strength transmitted by –NN– moiety of atr and hydroxyl mediators result in overall S = 0 (for 1) and 1/2 (for 3 and 4) spin ground states. These interesting results indicate that coordinative side group in the ternary metal ion-azolate-carboxylate system can be utilized to generate aesthetically pleasing building units and variably polytopic connectors, leading to differently extended superstructures and magnetic behavior.