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The concept of dielectric-laser acceleration (DLA) provides the highest gradients among breakdown-limited (nonplasma) particle accelerators and thus the potential of miniaturization. The implementation of a fully scalable electron accelerator on a microchip by two-dimensional alternating phase focusing (APF), which relies on homogeneous laser fields and external magnetic focusing in the third direction, was recently proposed. In this Letter, we generalize the APF for DLA scheme to 3D, such that stable beam transport and acceleration is