We study the formation of 53 galaxy cluster-size dark matter halos (M = 1014.0-14.76 M sun) formed within a pair of cosmological Λ cold dark matter N-body simulations, and track the accretion histories of cluster subhalos with masses large enough to host ~0.3 L * galaxies. By associating subhalos with cluster galaxies, we find the majority of galaxies in clusters experience no "preprocessing" in the group environment prior to their accretion into the cluster. On average, 70% of cluster galaxies fall into the cluster potential directly from the field, with no luminous companions in their host halos at the time of accretion; less than 12% are accreted as members of groups with five or more galaxies. Moreover, we find that cluster galaxies are significantly less likely to have experienced a merger in the recent past (lsim6 Gyr) than a field halo of the same mass. These results suggest that local cluster processes such as ram pressure stripping, galaxy harassment, or strangulation play the dominant role in explaining the difference between cluster and field populations at a fixed stellar mass, and that pre-evolution or past merging in the group environment is of secondary importance for setting cluster galaxy properties for most clusters. The accretion times for z = 0 cluster members are quite extended, with ~20% incorporated into the cluster halo more than 7 Gyr ago and ~20% within the last 2 Gyr. By comparing the observed morphological fractions in cluster and field populations, we estimate an approximate timescale for late-type to early-type transformation within the cluster environment to be ~6 Gyr.