  Radio-galaxy (RG) lobes contain relativistic electrons embedded in a tangled
magnetic field that produce, in addition to low-frequency synchrotron radio
emission, inverse-Compton scattering (ICS) of the cosmic microwave background
(CMB) photons. This produces a relativistic, non-thermal Sunyaev-Zel'dovich
effect (SZE). We study the spectral and spatial properties of the non-thermal
SZE in a sample of radio galaxies and make predictions for their detectability
in both the negative and the positive part of the SZE, with space experiments
like Planck, OLIMPO, and Herschel-SPIRE. These cover a wide range of
frequencies, from radio to sub-mm. We model the SZE in a general formalism that
is equivalent to the relativistic covariant one and describe the electron
population contained in the lobes of the radio galaxies with parameters derived
from their radio observations, namely, flux, spectral index, and spatial
extension. We further constrain the electron spectrum and the magnetic field of
the RG lobes using X-ray, gamma-ray, and microwave archival observations. We
determine the main spectral features of the SZE in RG lobes, namely, the
minimum, the crossover, and the maximum of the SZE. We show that these typical
spectral features fall in the frequency ranges probed by the available space
experiments. We provide the most reliable predictions for the amplitude and
spectral shape of the SZE in a sample of selected RGs with extended lobes. In
three of these objects, we also derive an estimate of the magnetic field in the
lobe at the muG level by combining radio (synchrotron) observations and X-ray
(ICS) observations. These data, together with the WMAP upper limits, set
constraints on the minimum momentum of the electrons residing in the RG lobes
and allow realistic predictions for the visibility of their SZE to be derived
with Planck, OLIMPO, and Herschel-SPIRE. [abridged]
