A new method for simulating the three-dimensional dynamics of charged energetic particles in very broadband noncompressive magnetic turbulence is introduced. All scales within the primary inertial range of the turbulence observed in the solar wind near 1 AU are now included for the independent computations of both the particle dynamics and the turbulent magnetic field lines (MFLs). While previous theories of resonant particle pitch-angle (PA) scattering and transport in interplanetary magnetic fields had favored interpreting the observed depletions in the electron PA distributions (PADs) around 90 • PA as evidence of poor scattering at low PA cosines, the computed particle dynamics reveal a very different reality. The MFL directions now vary on many scales, and the PADs are depleted around 90 • PA due to nonresonant filtering of the particles that propagate at too large an angle to the local magnetic field. Rather than being too weak, the scattering through 90 • PA is actually so strong that the particles (electrons and protons/ions) are reflected and trapped in the turbulent magnetic fields. While the low-frequency nonresonant turbulence produces ubiquitous magnetic traps that only let through particles with the most field-aligned velocities, higher-frequency near-gyroscale turbulence, when present, enhances particle transport by allowing the particles to navigate between magnetic traps. Finally, visualizing both particle trajectories and MFLs in the very same turbulence reveals a powerful tool for understanding the effects of the turbulent fields on the particle dynamics and cross-field transport. Some cross-field-line scattering, strongly amplified by MFL dispersal, results in a strong cross-field scattering of the particles. From this visualization, it also appears that near-gyroscale turbulence, previously known as gyroresonant turbulence, does not resonantly interact with the particles. The interaction between particles and fields at or near the gyroscale, though potentially strong, does not actually involve the periodic driving of a true resonance.