Chandra observations of a complete, flux-limited sample of 38 high-redshift (1 < z < 2), low-frequency-selected (and so unbiased in orientation) 3CRR radio sources are reported. The sample includes 21 quasars (=broad-line radio galaxies) and 17 narrow-line radio galaxies (NLRGs) with matched 178 MHz radio luminosity (log L_R(5 GHz) ~44-45). The quasars have high radio core fraction, high X-ray luminosities (log L_X ~45-46), and soft X-ray hardness ratios (HR ~–0.5) indicating low obscuration. The NLRGs have lower core fraction, lower apparent X-ray luminosities (log L_X ~43-45), and mostly hard X-ray hardness ratios (HR >0) indicating obscuration (N_H ~10^(22)-10^(24) cm^(–2)). These properties and the correlation between obscuration and radio core fraction are consistent with orientation-dependent obscuration as in unification models. About half the NLRGs have soft X-ray hardness ratios and/or a high [O III] emission line to X-ray luminosity ratio suggesting obscuration by Compton thick (CT) material so that scattered nuclear or extended X-ray emission dominates (as in NGC 1068). The ratios of unobscured to Compton-thin (10^(22) cm^(–2) < N_H(int) <1.5 × 10^(24) cm^(–2)) to CT (N_H(int) >1.5 × 10^(24) cm^(–2)) is 2.5:1.4:1 in this high-luminosity, radio-selected sample. The obscured fraction is 0.5, higher than is typically reported for active galactic nuclei at comparable luminosities from multi-wavelength surveys (0.1-0.3). Assuming random nuclear orientation, the unobscured half-opening angle of the disk/wind/torus structure is ~60° and the obscuring material covers 30°, ~12° of which is CT. The multi-wavelength properties reveal that many NLRGs have intrinsic absorption 10-1000 × higher than indicated by their X-ray hardness ratios, and their true L_X values are ~10-100× larger than the hardness-ratio absorption corrections would indicate.