This paper presents XMM-Newton and Chandra X-ray spectroscopy of ten flat-spectrum radio quasars (FSRQ) which are candidates to have an X-ray spectrum dominated by jet synchrotron emission. In all these FSRQ, which are less strongly relativistically beamed than blazars, a considerable contribution from a power-law component similar to that present in radio-quiet quasars is required to explain the observed X-ray fluxes and X-ray spectral slopes. And as in radio-quiet quasars, their relatively high optical/UV fluxes can be accounted for by a significant contribution from thermal accretion disk emission. The lack of success in finding radio quasars with synchrotron X-rays is attributed to the adopted selection criteria, which were based on the multiwavelength flux ratios of BL Lacertae (BL Lac) objects. A refined selection technique, which additionally involves radio imaging, is proposed to search for these important candidates for the Gamma Ray Large Area Space Telescope (GLAST). On the other hand, the discovered FSRQ with their strong accretion disk signatures are expected to be important probes for studies of the poorly known accretion disk - jet connection.