Molecular dielectric materials require ostensibly conflicting requirements of high polarizability and low conductivity. As previous efforts towards molecular insulators focused on saturated molecules, it remains an open question whether pi- and sigma-transport can be simultaneously suppressed in conjugated systems. Here, we demonstrate that there are conjugated molecules where the sigma-transmission is suppressed by destructive sigma-interference, while the pi-transmission can be suppressed by a localized disruption of conjugation. Using density functional theory, we study the Landauer transmission and ballistic current density, which allow us to determine how the transmission is affected by various structural changes in the molecule. We find that in para-linked oligophenyl rings the sigma-transmission can be suppressed by changing the remaining hydrogens to methyl groups due to the inherent gauche-like structure of the carbon backbone within a benzene ring, similar to what was previously seen in saturated systems. At the same time, the methyl groups fulfil a dual purpose as they modulate the twist angle between neighboring phenyl rings. When neighboring rings are orthogonal to each other, the transmission through both pi- and sigma-systems is effectively suppressed. Alternatively, breaking conjugation in a single phenyl ring by saturating two carbons atoms with two methyl substituents on each carbon, results in suppressed pi- and sigma-transport independent of dihedral angle. These two strategies demonstrate that methyl-substituted oligophenyls are promising candidates for the development of molecular dielectric materials.