Some basic properties of collisionless, trapped-electron mode turbulence in tokamaks are investigated by means of massively parallel gyrokinetic Vlasov simulations. In particular, the spatial structure and wave number spectra of various fluctuating plasma quantities are presented and discussed. An analysis of several cross phase relations supports the view that the transport-dominating scales may be interpreted in terms of remnants of linear modes. In a few test cases, zonal flows are artificially suppressed, demonstrating that their influence on the transport level is small. Finally, the dependence of the latter on several plasma parameters is studied.