This paper analyses the radio properties of a subsample of optically obscured (R>=25.5) galaxies observed at 24um by the Spitzer Space Telescope within the First Look Survey. Ninety-six F24um>=0.35mJy objects out of 510 are found to have a radio counterpart at 1.4GHz, 610MHz or at both frequencies, respectively, down to ~40 and ~200uJy. IRAC photometry sets the majority of them in the redshift interval z~=[1-3] and allows for a broad distinction between active galactic nucleus (AGN) dominated galaxies (~47 per cent of the radio-identified sample) and systems powered by intense star formation (~13 per cent), the remaining objects being impossible to classify. The percentage of radio identifications is a strong function of 24-um flux: almost all sources brighter than F_24um_~2mJy are endowed with a radio flux at both 1.4GHz and 610MHz, while this fraction drastically decreases by lowering the 24-um flux level.