We developed a three-parameter Born-von Karman type model of lattice dynamics for nanolayers with diamond crystal lattice. In the framework of this model we theoretically investigated the phonon energy spectra and in-plane thermal conductivity of nanometer-thick silicon nanolayers taking into account two-dimensional density of phonon states and phonon modes quantization. We demonstrated that optical phonons practically do not participate in the heat transport in the considered nanolayers and their contribution to the thermal conductivity constitutes only a few percent. We compared our results with those obtained using the Face-Centered Cubic Cell model of lattice dynamics and found that both models give qualitatively similar results. The calculated temperature dependences of the phonon thermal conductivity are found in a good agreement with available experimental data for 20-nm-thick and 30-nm-thick silicon nanolayers. The proposed theoretical model can be useful for description of the quantized phonon modes and thermal conductivity in the nanolayers or nanowires embedded in the coating layers with different acoustic properties.