Combining dielectric spectroscopy and neutron scattering data for hydrated lysozyme powders, we were able to identify several relaxation processes and follow protein dynamics at different hydration levels over a broad frequency and temperature range. We ascribe the main dielectric process to protein's structural relaxation coupled to hydration water and the slowest dielectric process to a larger scale protein's motions. Both relaxations exhibit a smooth, slightly super-Arrhenius temperature dependence between 300 and 180 K. The temperature dependence of the slowest process follows the main dielectric relaxation, emphasizing that the same friction mechanism might control both processes. No signs of a proposed sharp fragile-to-strong crossover at T similar to 220 K are observed in temperature dependences of these processes. Both processes show strong dependence on hydration: the main dielectric process slows down by six orders with a decrease in hydration from h similar to 0.37 (grams of water per grams of protein) to h similar to 0.05. The slowest process shows even stronger dependence on hydration. The third (fastest) dielectric relaxation process has been detected only in samples with high hydration (h similar to 0.3 and higher). We ascribe it to a secondary relaxation of hydration water. The mechanism of the protein dynamic transition and a general picture of the protein dynamics are discussed.