We present the results of an investigation into the X-ray properties of radio-intermediate and radio-loud quasars (RIQs and RLQs, respectively). We combine large, modern optical (e.g., SDSS) and radio (e.g., FIRST) surveys with archival X-ray data from Chandra, XMM-Newton, and ROSAT to generate an optically selected sample that includes 188 RIQs and 603 RLQs. This sample is constructed independently of X-ray properties but has a high X-ray detection rate (85%); it provides broad and dense coverage of the l-z plane, including at high redshifts (22% of objects have z=2-5), and it extends to high radio-loudness values (33% of objects have R*=3-5, using logarithmic units). We measure the "excess" X-ray luminosity of RIQs and RLQs relative to radio-quiet quasars (RQQs) as a function of radio loudness and luminosity, and parameterize the X-ray luminosity of RIQs and RLQs both as a function of optical/UV luminosity and also as a joint function of optical/UV and radio luminosity. RIQs are only modestly X-ray bright relative to RQQs; it is only at high values of radio loudness (R*>~3.5) and radio luminosity that RLQs become strongly X-ray bright.