The 1D coordination polymer (CP) [(Me2S)3{Cu2(μ-I)2}]n (1) is formed when CuI reacts with SMe2 in n-heptane, whereas in acetonitrile (MeCN), the reaction forms exclusively the 2D CP [(Me2S)3{Cu4(μ-I)4}]n (2) containing "flower-basket" Cu4I4 units. The reaction product of CuI with MeSEt is also solvent-dependent, where the 1D polymer [(MeSEt)2{Cu4(μ3-I)2(μ2-I)2}(MeCN)2]n (3) containing "stepped-cubane" Cu4I4 units is isolated in MeCN. In contrast, the reaction in n-heptane affords the 1D CP [(MeSEt)3{Cu4(μ3-I)4}]n (4) containing "closed-cubane" Cu4I4 clusters. The reaction of MeSPr with CuI provides the structurally related 1D CP [(MeSPr)3{Cu4(μ3-I)4}]n (5), for which the X-ray structure has been determined at 115, 155, 195, 235, and 275 K, addressing the evolution of the metric parameters. Similarly to 4 and the previously reported CP [(Et2S)3{Cu4(μ3-I)4}]n (Inorg. Chem. 2010, 49, 5834), the 1D chain is built upon closed cubanes Cu4(μ3-I)4 as secondary building units (SBUs) interconnected via μ-MeSPr ligands. The 0D tetranuclear clusters [(L)4{Cu4(μ3-I)4}] [L = EtSPr (6), Pr2S (7)] respectively result from the reaction of CuI with EtSPr and n-Pr2S. With i-Pr2S, the octanuclear cluster [(i-Pr2S)6{Cu8(μ3-I)3}(μ4-I)2}] (8) is formed. An X-ray study has also been performed at five different temperatures for the 2D polymer [(Cu3Br3)(MeSEt)3]n (9) formed from the reaction between CuBr and MeSEt in heptane. The unprecedented framework of 9 consists of layers with alternating Cu(μ2-Br)2Cu rhomboids, which are connected through two μ-MeSEt ligands to tetranuclear open-cubane Cu4Br4 SBUs. MeSPr forms with CuBr in heptane the 1D CP [(Cu3Br3)(MeSPr)3]n (10), which is converted to a 2D metal-organic framework [(Cu5Br5)(μ2-MeSPr)3]n (11) incorporating pentanuclear [(Cu5(μ4-Br)(μ2-Br)] SBUs when recrystallized in MeCN. The thermal stability and photophysical properties of these materials are also reported.