  We describe photonic crystal microcavities with very strong light-matter
interaction to realize room-temperature, equilibrium, exciton-polariton
Bose-Einstein condensation (BEC). This is achieved through a careful balance
between strong light-trapping in a photonic band gap (PBG) and large exciton
density enabled by a multiple quantum-well (QW) structure with moderate
dielectric constant. This enables the formation of long-lived, dense 10~$\mu$m
- 1~cm scale cloud of exciton-polaritons with vacuum Rabi splitting (VRS) that
is roughly 7\% of the bare exciton recombination energy. We introduce a
woodpile photonic crystal made of Cd$_{0.6}$Mg$_{0.4}$Te with a 3D PBG of 9.2\%
(gap to central frequency ratio) that strongly focuses a planar guided optical
field on CdTe QWs in the cavity. For 3~nm QWs with 5~nm barrier width the
exciton-photon coupling can be as large as $\hbar\Ome=$55~meV (i.e., vacuum
Rabi splitting $2\hbar\Ome=$110~meV). The exciton recombination energy of
1.65~eV corresponds to an optical wavelength of 750~nm. For $N=$106 QWs
embedded in the cavity the collective exciton-photon coupling per QW,
$\hbar\Ome/\sqrt{N}=5.4$~meV, is much larger than state-of-the-art value of
3.3~meV, for CdTe Fabry-P\'erot microcavity. The maximum BEC temperature is
limited by the depth of the dispersion minimum for the lower polariton branch,
over which the polariton has a small effective mass $\sim 10^{-5}m_0$ where
$m_0$ is the electron mass in vacuum. By detuning the bare exciton
recombination energy above the planar guided optical mode, a larger dispersion
depth is achieved, enabling room-temperature BEC.
