We present a uniform and systematic analysis of the 0.6-10 keV X-ray spectra of radio-loud active galactic nuclei (AGNs) observed by ASCA. The sample, which is not statistically complete, includes 10 broad-line radio galaxies (BLRGs), five radio-loud quasars (QSRs), nine narrow-line radio galaxies (NLRGs), and 10 radio galaxies (RGs) of mixed FR I and FR II types. For several sources the ASCA data are presented here for the first time. The exposure times of the observations and the fluxes of the objects vary over a wide range; as a result, so does the signal-to-noise ratio of the individual X-ray spectra. At soft X-rays, about 50% of NLRGs and 100% of RGs exhibit thermal plasma emission components, with bimodal distributions of temperatures and luminosities. This indicates that the emission in such an object arises in hot gas either in a surrounding cluster or loose group or in a hot corona, consistent with previous ROSAT and optical results. At energies above 2 keV, a hard power-law component (photon index Γ ~ 1.7-1.8) is detected in 90% of cases. The power-law photon indices and luminosities in BLRGs, QSRs, and NLRGs are similar. This is consistent with simple orientation-based unification schemes for lobe-dominated radio-loud sources in which BLRGs, QSRs, and NLRGs harbor the same type of central engine. Moreover, excess cold absorption in the range 1021-1024 cm-2 is detected in most (but not all) NLRGs, consistent with absorption by obscuring tori, as postulated by unification scenarios. The ASCA data provide initial evidence that the immediate gaseous environment of the X-ray source of BLRGs may be different than in Seyfert 1 galaxies: absorption edges of ionized oxygen, common in the latter, are detected in only one BLRG. Instead we detect large columns of cold gas in a fraction (~44%-60%) of BLRGs and QSRs, comparable to the columns detected in NLRGs, which is puzzling. This difference hints at different physical and/or geometrical properties of the medium around the X-ray source in radio-loud AGNs compared to their radio-quiet counterparts, properties that can be explored further with future X-ray observations. For the full sample, the nuclear X-ray luminosity is correlated with the luminosity of the [O III] emission line, the FIR emission at 12 μm, and the lobe radio power at 5 GHz. The Fe Kα line is detected in 50% of BLRGs and in one QSR, with a large range of intrinsic widths and equivalent widths. In the handful of NLRGs where it is detected, the line is generally unresolved. Comparing the average power-law photon indices of the various classes of radio-loud AGNs to their radio-quiet counterparts from the literature, we find only a weak indication that the ASCA 2-10 keV spectra of BLRGs are flatter than those of Seyfert 1 galaxies of comparable X-ray luminosity. This result is at odds with evidence from samples studied by other authors suggesting that radio-loud AGNs have flatter spectra than radio-quiet ones. Rather, it supports the idea that a beamed synchrotron self-Compton component related to the radio source (jet) is responsible for the flatter slopes in those radio-loud AGNs. We argue that, because of the way those samples were constructed, beamed X-ray emission from the radio jets probably contributed to the observed X-ray spectra. The sample studied here includes six weak-line radio galaxies (WLRGs), powerful radio galaxies characterized by [O III] 4569 and 5007 A of unusually low luminosity and by unusually high [O II]/[O III] line ratios. The ASCA spectra of WLRGs can be generally decomposed into a soft thermal component with kT ~ 1 keV, plus a hard component, described either by a flat (Γ = 1.5) absorbed power law or by a very hot (kT ~ 100 keV) thermal bremsstrahlung model. Their intrinsic luminosities are in the range L2-10 keV ~ 1040-1042 ergs s-1, 2 orders of magnitude lower than in other sources in our sample. If the hard X-ray emission is attributed to a low-luminosity AGN, an interesting possibility is that WLRGs represent an extreme population of radio galaxies in which the central black hole is accreting at a rate well below the Eddington rate.