Metamaterials (MMs) and photonic crystals (PhCs) exhibiting artificially engineered physical properties have been widely studied in the past decade. However, abnormal properties could only be proposed under a limited range of polarization and directions in most of the previous studies. It is still a challenge to realize an isotropic artificial material with multiple exotic electromagnetic properties. Here we report a three-dimensional metacrystal supporting full polarization and omni-directional incidence. The center-symmetric unit cell consists of non-resonant closed metallic loops on each surface of the dielectric cube. With the cross-scale dispersion engineering, the metacrystal can exhibit an isotropic diamagnetic response and an all-angle negative index of refraction simultaneously at the opposite sides of the MM-PhC transition region. An additional numerical analysis shows the good performance in terahertz and mid-infrared frequencies, which indicates its potential applications on multi-functional optical components with wide polarization-and-direction allowance.