Crystal and electronic structures and physical properties of the molecular conductors Me(4)Z [Pd(L)(2)](2) (Z = N, P, As, and Sb; L = dmit and dmise) are described. Every salt has very similar crystal structure with "solid-crossing" columns. The behavior of resistivity under pressure depends on the Se-substitution and the choice of the cation. Tight-binding band calculations indicate that these salts form a unique two-band system. The narrow and two-dimensional HOMO (anti-bonding) band is thought to form the half-filled conduction band. Magnetic properties suggest that the insulating state under ambient pressure comes from the strong electron-electron correlation. The one-dimensional LUMO (bonding) band is located below the HOMO band. The application of pressure induces an overlap of these two bands and changes the band filling, which brings forth the metallic state. The system under higher pressure, however, shows nonmetallic behavior probably due to the one-dimensional instability associated with the LUMO band. Se-substitution and cation effects are observed in the band structure and the shape of Fermi surface. Our results suggest an interrelation between the correlation effect and the dimensionality of the electronic structure.