We demonstrate that vertically aligned InN nanorods can be grown on Si(111) by plasma-assisted molecular-beam epitaxy. Detailed structural characterization indicates that individual nanorods are wurtzite InN single crystals with the growth direction along the c axis. Near-infrared photoluminescence (PL) from InN nanorods can be clearly observed at room temperature. However, in comparison to the InN epitaxial films, the PL efficiency is significantly lower. Moreover, the variable-temperature PL measurements of InN nanorods exhibit anomalous temperature effects. We propose that these unusual PL properties are results of considerable structural disorder (especially for the low-temperature grown InN nanorods) and strong surface electron accumulation effects.