  Upper limits from the current generation of interferometers targeting the
21-cm signal from high redshifts have recently begun to rule out physically
realistic, though still extreme, models of the Epoch of Reionization (EoR).
While inferring the detailed properties of the first galaxies is one of the
most important motivations for measuring the high-$z$ 21-cm signal, they can
also provide useful constraints on the properties of the intergalactic medium
(IGM). Motivated by this, we build a simple, phenomenological model for the
21-cm power spectrum that works directly in terms of IGM properties, which
bypasses the computationally expensive 3-D semi-numerical modeling generally
employed in inference pipelines and avoids explicit assumptions about galaxy
properties. The key simplifying assumptions are that (i) the ionization field
is binary, and composed of spherical bubbles with an abundance described well
by a parametric bubble size distribution, and (ii) that the spin temperature of
the ``bulk'' IGM outside bubbles is uniform. Despite the simplicity of the
model, the mean ionized fraction and spin temperature of the IGM recovered from
mock 21-cm power spectra generated with \textsc{21cmfast} are generally in good
agreement with the true input values. This suggests that it is possible to
obtain comparable constraints on the IGM using models with very different
assumptions, parameters, and priors. Our approach will thus be complementary to
semi-numerical models as upper limits continue to improve in the coming years.
