  We explore the possibility of detecting HI 21cm absorption by the neutral
intergalactic medium (IGM) toward very high redshift radio sources. The epoch
considered is between the time when the first ionizing sources form and when
the bulk of the neutral IGM becomes ionized. Due to the extreme Ly-alpha
opacities of the neutral IGM, objects within this 'gray age' can only be
observed at wavelengths longer than about 1micron. We use the latest
simulations of the evolution of the IGM in the context of LambdaCDM structure
formation models constrained by observations of the highest redshift QSOs to
predict the optical depth as a function of frequency of the neutral IGM due to
the HI 21cm line. We then simulate radio spectra assuming observational
parameters for future large area radio telescopes. These spectra show that HI
21cm absorption studies could be a powerful probe of the rich structure of the
neutral IGM prior to the epoch of reionization, including 1% absorption by the
mean neutral IGM, plus deeper, narrow lines (5% and a few km/s). Most of the
variations in transmissivity are due to the mild density inhomogeneities with
typical values of the cosmic overdensity of about 10, precisely the structures
that at later times give rise to the Ly-alpha forest. We also consider
sensitivity limits and the evolution of radio source populations, and conclude
that it is reasonable to hypothesize the existence of an adequate number of
high-z radio sources against which such absorption studies could be performed.
