We report on a study of interstellar turbulence driven by both correlated and isolated supernova explosions. We use three-dimensional hydrodynamic models of a vertically stratified interstellar medium run with the adaptive mesh refinement code Flash at a maximum resolution of 2 pc, with a grid size of 0.5 × 0.5 × 10 kpc. Cold dense clouds form even in the absence of self-gravity due to the collective action of thermal instability and supersonic turbulence. Studying these clouds, we show that it can be misleading to predict physical properties such as the star formation rate or the stellar initial mass function using numerical simulations that do not include self-gravity of the gas. Even if all the gas in turbulently Jeans unstable regions in our simulation is assumed to collapse and form stars in local freefall times, the resulting total collapse rate is significantly lower than the value consistent with the input supernova rate. The amount of mass available for collapse depends on scale, suggesting a simple translation from the density PDF to the stellar IMF may be questionable. Even though the supernova-driven turbulence does produce compressed clouds, it also opposes global collapse. The net effect of supernova-driven turbulence is to inhibit star formation globally by decreasing the amount of mass unstable to gravitational collapse.