Plastic dielectric materials are required in radiation and rare event physics detectors as structural parts, insulators, seals, light reflectors, and in connectors. In order to observe events of interest rather than counting the radioactivity of the detector itself, these materials must be of high radiopurity. Using a mass spectrometric method, a number of solid plastic materials and parts were analyzed for mass levels of contaminants 232Th and 238U, which were used as an assessment of radiopurity. Sample preparation began with dry ashing the solid subsamples in very low mass containers of ultra low background electroformed copper (ULB EF-Cu) foil that had been cut and folded into boats serving as crucibles. Final determinations were carried out by inductively coupled plasma mass spectrometry (ICP-MS) with liquid nebulization sample solution introduction. Sample sizes were in the 25 mg to 1.25 g mass, and analyses could be completed in less than two days. Detection limits to pg/g (i.e, parts per trillion) levels are obtained and converted to radioactivity units μBq/kg of material. Determinations on poly(methyl methacrylate) (PMMA) gave values for 238U below 10 μBq/kg, while values for poly(chlorotrifluoroethylene) (PCTFE) rod stock material or acid-leached machined PCTFE parts were at or below 50 μBq/kg in 238U. Results for solid Purad poly(vinylidene fluoride) (PVDF) material are also reported at ca 5 mBq/kg of 238U, although PVDF Solef 1010 raw material in powder and pellet form has been previously reported as low as 25 μBq/kg, and 200 μBq/kg 238U, respectively. An analysis of a polyetherimide (PEI) material, sold as ULTEM, indicated promising radiopurity with a found value of 30 μBq/kg for 238U. A variety of solid dielectric polymer materials were found with radiopurities at less than a mBq/kg 238U and 232Th.