Exploring the ICM power spectrum can help us to probe the physics of galaxy clusters. Using high-resolution 3D plasma simulations, we study the statistics of the velocity field and its relation with the thermodynamic perturbations. The normalization of the ICM spectrum (density, entropy, or pressure) is linearly tied to the level of large-scale motions, which excite both gravity and sound waves due to stratification. For low 3D Mach number M~0.25, gravity waves mainly drive entropy perturbations, traced by preferentially tangential turbulence. For M>0.5, sound waves start to significantly contribute, passing the leading role to compressive pressure fluctuations, associated with isotropic (or slightly radial) turbulence. Density and temperature fluctuations are then characterized by the dominant process: isobaric (low M), adiabatic (high M), or isothermal (strong conduction). Most clusters reside in the intermediate regime, showing a mixture of gravity and sound waves, hence drifting towards isotropic velocities. Remarkably, regardless of the regime, the variance of density perturbations is comparable to the 1D Mach number. This linear relation allows to easily convert between gas motions and ICM perturbations, which can be exploited by Chandra, XMM data and by the forthcoming Astro-H. At intermediate and small scales (10-100 kpc), the turbulent velocities develop a Kolmogorov cascade. The thermodynamic perturbations act as effective tracers of the velocity field, broadly consistent with the Kolmogorov-Obukhov-Corrsin advection theory. Thermal conduction acts to damp the gas fluctuations, washing out the filamentary structures and steepening the spectrum, while leaving unaltered the velocity cascade. The ratio of the velocity and density spectrum thus inverts the downtrend shown by the non-diffusive models, allowing to probe the presence of significant conductivity in the ICM. ; Comment: Accepted by A&A; 15 pages, 10 figures; added insights and references - thank you for the positive feedback