  We describe a cryogenic cavity-optomechanical system that combines Si3N4
membranes with a mechanically-rigid Fabry-Perot cavity. The extremely high
quality-factor frequency products of the membranes allow us to cool a MHz
mechanical mode to a phonon occupation of less than 10, starting at a bath
temperature of 5 kelvin. We show that even at cold temperatures
thermally-occupied mechanical modes of the cavity elements can be a limitation,
and we discuss methods to reduce these effects sufficiently to achieve ground
state cooling. This promising new platform should have versatile uses for
hybrid devices and searches for radiation pressure shot noise.
