(Abridged) The hot gas in the IGM produces anisotropies in the Cosmic Microwave Background (CMB) through the thermal Sunyaev-Zel'dovich (SZ) effect. The SZ effect is a powerful probe of large-scale structure in the universe and must be carefully subtracted from measurements of the primary CMB anisotropies. We use moving-mesh hydrodynamical simulations to study the 3-dimensional statistics of the gas, and compute the mean comptonization parameter and the angular power spectrum of the SZ

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... ons, for different cosmologies. We compare these results with predictions using the Press-Schechter formalism. We find that the two methods agree approximately, but differ in details. We discuss this discrepancy, and show that resolution limits the reliability of our results to the 200<l<2000 range. For cluster- normalized CDM models, the SZ power spectrum is comparable to the primordial power spectrum around l=2000. We show that groups and filaments (kT<5 keV) contribute about 50% of the SZ power spectrum at l=500. About half of the SZ power spectrum on these scales is produced at redshifts z<0.1, and can thus be detected and removed using existing catalogs of galaxies and X-ray clusters. We discuss the implications of these results for the future MAP and Planck Surveyor missions.