Distant radio galaxies are powerful radio sources and have strong emission lines. The emission-line regions are quite extended, often 100 kpc along the major axis, and are clumpy, elongated, and aligned with the radio axis. In addition, the emission-line luminosity is roughly proportional to the radio power. Energy transferred from relatively low-energy relativistic electrons to the ambient gas may contribute significantly to the emission-line luminosity of the galaxy. Interactions between relativistic electrons and ambient gas heats the gas, which then radiates line emission. The line luminosity is expected to be proportional to the radio power in this model, in agreement with the observations. The observed normalization requires that cold gas with an average density between about 5 x (10(-3) to 10(-1)) cm-3 be present along the radio bridge of the radio source, indicating a total gas mass within a spherical volume with a 50 kpc radius about the radio galaxy of about 5 x (10(10) to 10(12)) M.. Relativistic electrons lose energy and cool down as they heat the ambient gas. Relatively cool relativistic electrons inverse Compton scatter microwave-background photons to optical and ultraviolet energies, while higher energy relativistic electrons inverse Compton scatter microwave photons to X-ray energies. Thus when interactions between relativistic electrons and ambient gas are an important source of power for the line emission, interactions between relatively cool relativistic electrons and microwave photons are likely to produce a local source of ultraviolet radiation that will affect the emission-line ratios.