We have performed a first principles density functional theory (DFT) calculations to study the thermal conductivity in ZnO nanotubes, ZnO nanowires, and Si/Ge shell-core nanowires. We found the equilibrium configuration and the electric band structure of each nanostructure using DFT, the interatomic force constants and the phonon dispersion relations were calculated using DFPT as implemented in Quantum Espresso. In order to fundamentally understand the effect of atomic arrangements, we calculated the phonon conductance in a ballistic approach using a Green's function method. All ZnO nanostructures studied exhibit semiconducting behavior, with direct bandgap at the Gamma point. The calculated values for the bandgaps were larger than the value of the bandgap of the bulk ZnO. We were able to identify phonon modes in which the motion of Zn atoms is significant when it is compared with the motion of oxygen atoms. The thermal conductivity depends on the diameter of the nanowires and nanotubes and it is dramatically affected when the nanowire or nanotube is doped with Ga. For Si/Ge nanowires, the slope and the curvature of acoustic modes in the phonon dispersion relation increases when the diameter increases. For nanowires with the same number of atoms, the slope and curvature of acoustic modes depends on the concentration of Si atoms. We were able to identify phonon modes in which the motion of core atoms is significant when it is compared with motion of atoms on the nanowire's shell. The thermal conductivity in these nanostructures depends on the nanowire's diameter and on the Si atoms concentration.