A series of quasi-one-dimensional molecular conductors (DMEDO-TTF)(2)XF(6) (X = P, As, and Sb), where DMEDO-TTF is dimethyl(ethylenedioxy)tetrathiafulvalene, undergo characteristic structural transitions in the range of 130-195 K for the PF(6) salt and 222-242 K for the AsF(6) salt. The dramatic structural transition is induced by the order of the ethylenedioxy moiety, and the resulting anion rotation leads to the reconstruction of the H···F interaction between the methyl groups and the anions. The unique hydrogen bonds play a crucial role in the transition. As a result, the molecular packing is rearranged entirely; the high-temperature molecular stacks with an ordinary quasi-triangular molecular network transforms to a quasi-square-like network, which has never been observed among organic conductors. Nonetheless, the low-temperature phase exhibits a good metallic conductivity as well, so the transition is a metal-metal (MM) transition. The resistivity measured along the perpendicular direction to the conducting ac-plane (ρ(⊥)) and the calculation of the Fermi surface demonstrate that the high-temperature metal phase is a one-dimensional metal, whereas the low-temperature metal phase has considerable interchain interaction. In the SbF(6) salt, a similar structural transition takes place around 370 K, so that the quasi-square-like lattice is realized even at room temperature. Despite the largely different MM transition temperatures, all these salts undergo metal-insulator (MI) transitions approximately at the same temperature of 50 K. The low-temperature insulator phase is nonmagnetic, and the reflectance spectra suggest the presence of charge disproportionation with small charge difference (0.14).