We present a detailed computational study of the assembly of protostellar disks and massive stars in molecular clouds with supersonic turbulence. We follow the evolution of large scale filamentary structures in a cluster-forming clump down to protostellar length scales by means of very highly resolved, 3D adaptive mesh refined (AMR) simulations, and show how accretion disks and massive stars form in such environments. We find that an initially elongated cloud core which has a slight spin from oblique shocks collapses first to a filament and later develops a turbulent disk close to the center of the filament. The continued large scale flow that shocks with the filament maintains the high density and pressure within it. Material within the cooling filament undergoes gravitational collapse and an outside-in assembly of a massive protostar. Our simulations show that very high mass accretion rates of up to 10^-2 Msol/yr and high, supersonic, infall velocities result from such filamentary accretion. Accretion at these rates is higher by an order of magnitude than those found in semi-analytic studies, and can quench the radiation field of a growing massive young star.Our simulations include a comprehensive set of the important chemical and radiative processes such as cooling by molecular line emission, gas-dust interaction, and radiative diffusion in the optical thick regime, as well as H2 formation and dissociation. Therefore, we are able to probe, for the first time, the relevant physical phenomena on all scales from those characterizing the clump down to protostellar core.