  We demonstrate coherent Doppler-free three-photon excitation of the
$^{1}S_{0}$$\leftrightarrow$$^{3}P_{0}$ optical clock transition and the
$^{1}S_{0}$$\leftrightarrow$$^{3}P_{1}$ intercombination transition in
free-space thermal clouds of $^{88}$Sr atoms. By appropriate orientation of the
wavevectors of three lasers incident on the atoms, the first-order Doppler
shift can be eliminated for all velocity classes. Three-photon excitation of
the $^{1}S_{0}$$\leftrightarrow$$^{3}P_{1}$ transition enables high-contrast
Ramsey spectroscopy with interrogation times comparable to the 21$\mu$s natural
lifetime using a single near-resonant laser source. Three-photon spectroscopy
on the $^{1}S_{0}$$\leftrightarrow$$^{3}P_{0}$ clock transition, using only
laser frequencies nearly resonant with the
$^{1}S_{0}$$\leftrightarrow$$^{3}P_{0}$ and
$^{1}S_{0}$$\leftrightarrow$$^{3}P_{1}$ transitions, enables a reduction in
Doppler broadening by two orders of magnitude and a corresponding $\sim470$Hz
linewidth without a confining potential.
