Rapid accretion of cold gas plays a crucial role in getting gas into galaxies. It has been suggested that this accretion proceeds along narrow streams that might also directly drive the turbulence in galactic gas, dynamical disturbances, and bulge formation. In cosmological simulations, however, it is impossible to isolate and hence disentangle the effect of accretion from internal instabilities and mergers. Moreover, in most cosmological simulations, the phase structure and turbulence in the ISM arising from stellar feedback are treated in a sub-grid manner, so that feedback cannot generate ISM turbulence. In this paper we therefore test the effects of cold streams in extremely high-resolution simulations of otherwise isolated galaxy disks using detailed models for star formation and feedback; we then include or exclude mock cold flows falling onto the galaxies with accretion rates, velocities and geometry set to maximize their effect on the disk. We find: (1) Turbulent velocity dispersions in gas disks are identical with or without the cold flow; the energy injected by the flow is dissipated where it meets the disk. (2) In runs without stellar feedback, the presence of a cold flow has essentially no effect on runaway local collapse, resulting in star formation rates (SFRs) that are far too large. (3) Disks in runs with feedback and cold flows have higher SFRs, but only insofar as they have more gas. (4) Because flows are extended relative to the disk, they do not trigger strong resonant responses and so induce weak morphological perturbation (bulge formation via instabilities is not accelerated). (5) However, flows can thicken the disk by direct contribution of out-of-plane streams. We conclude that while inflows are critical over cosmological timescales to determine the supply and angular momentum of gas disks, they have weak instantaneous dynamical effects on galaxies.