A microscopic theory is developed of finite-size effects in sintered metallic powders. In particular a model is constructed for the low frequency vibrational modes. It is argued that the bonding between atoms on adjacent surfaces of neighboring grains is extremely weak, 10 ^-3 -10 ^-4 of its bulk value. Choosing the physical parameters accordingly, a constant density of "shaking box" states can be obtained. However, this behavior is not universal. A more complex density of states arises from slightly different choices of physical parameters and leads to a variety of temperature dependences for the sinter/liquid He Kapitza conductance at ultralow temperatures.