We investigate the assembly histories 56 of galaxy clusters using a pair of cosmological LCDM N-body simulations. We track infalling galaxy subhalos with masses large enough to host 0.1 L* galaxies , and find that the majority of cluster member galaxies experienced no ``pre-processing'' in the group environment prior to their accretion into the cluster. Only 14 % were accreted from "group" environments with five or more companions and approximately 65 % of our cluster members were accreted directly from the field, with no luminous companions in their host halos. Moreover, galaxies in our clusters are significantly {em less} likely to have had a recent (< 6 Gyr) large (>1/4) merger event than a field halo of the same mass. These results suggest that local, cluster processes like 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. The accretion times for z=0 cluster members are quite extended, with 20% incorporated into the cluster halo more than 7 Gyr ago and 45 % accreted within the last 3 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.