We have used pulsed-laser deposition, following a specific sequence of heating and cooling phases, to grow ZnO nanorods on ZnO buffer/Si (100) substrates, in a 600 mT oxygen ambient, without catalyst. In these conditions, the nanorods preferentially self-organize in the form of vertically aligned, core/shell structures. X-ray diffraction analyses, obtained from 2θ-ω and pole figure scans, shows a crystalline (wurtzite) ZnO deposit with uniform c-axis orientation normal to the substrate. Field emission SEM, TEM, HR-TEM and selective area electron diffraction (SAED) studies revealed that the nanorods have a crystalline core and an amorphous shell. The low-temperature (13 K) photoluminescence featured a strong I6 (3.36 eV) line emission, structured green band emission and a hitherto unreported broad emission at 3.331 eV. Further studies on the 3.331 eV band showed the involvement of deeply-bound excitonic constituents in a single electron-hole recombination. The body of structural data suggests that the 3.331 eV emission can be linked to the range of defects associated with the unique crystalline ZnO/amorphous ZnO core/shell structure of the nanorods. The relevance of the work is discussed in the context of the current production methods of core/shell nanorods and their domains of application.