We present the results of a detailed adiabatic survey of the pulsation properties of evolutionary models of pulsating DA white dwarfs. Pulsation periods, kinetic energies, and first-order rotation coefficients were calculated for all g-modes with l less-than-or-equal-to 3 in the period window 0-1000 s. The survey was carried out with the help of a new, high-performance adiabatic pulsation code based on the Galerkin finite-element method of weighted residuals. A very large volume of parameter space was explored in order to obtain a complete picture of the fundamental asteroseismological properties of DA white dwarfs. More than 1000 different compositionally stratified equilibrium models were considered in this exploration. The effects of variations in the hydrogen layer mass, the helium layer mass, the convective efficiency, the treatment of the composition transition zones, the total mass, and the effective temperature on the period structure were systematically investigated. We discuss the trends observed in the exact results and compare them with the predictions of an asymptotic model which we have developed previously. We emphasize the key role played by chemical stratification (particularly the H/He transition zone) in the pulsation characteristics of the models. The data bank described in this paper will be particularly useful for interpreting the period spectra and the rates of period changes being observed in ZZ Ceti stars.