Molecular adsorption of methanol on Cu(111), Cu(100), and Cu(110) surfaces in the 90-200 K temperature range is studied by a combination of infrared (IR) spectroscopy, thermal desorption analysis, and low-energy electron diffraction. Our results reveal the occurrence of a structural transformation in the H-bonded methanol assembly following heating from 90 to 120 K and demonstrate the effect of Cu surface orientation on the produced H-bonded structures, as well as thermal stability and ordering. At 90 K, the IR spectra indicate that similar H-bonded structures, presumably linear chains, are formed on all three surfaces. However, on Cu(111) the chains are assembled in ordered domains, whereas on Cu(100) and Cu(110) the chains are disordered. Heating to 120-130 K causes prominent changes in the IR spectrum of methanol on all surfaces, but there are significant differences between Cu(111) and the two other surfaces. We believe that such differences originate from different H-bonded methanol structures obtained on each surface after thermal annealing. On Cu(111) we suggest that cyclic structures (probably hexamers) are prevalent, whereas on Cu(100) and Cu(110) both cyclic and chain structures may coexist. The H-bonded structures produced at 120 K exhibit no long-range order on all three surfaces and show stronger adsorption (higher desorption temperate) on Cu(111). The higher stability and ordering (only at low temperatures) of adsorbed methanol on Cu(111) are attributed to the matching between the geometry of H-bonded methanol clusters and the surface structure.