A charged particle emits bremsstrahlung while traversing matter. We calculate the radiation cross section for bare heavy ions penetrating amorphous materials and single crystals at highly relativistic energies. The main component originates in scattering of the virtual photons of screened target nuclei on the projectile. It appears at, approximately, 2γ times the energy of the giant dipole resonance of the projectile, approximately 25γ MeV for a lead nucleus (γ≡E/Mc2, where E and M denote the projectile energy and mass). The emission pertains to relatively close impacts, with impact parameters ranging to, at maximum, the screening radius of the target atoms. As a result, the main bremsstrahlung component shows channeling dips, that is, dips in yield upon variation of the incidence angle to major crystallographic directions of a single crystal. The minimum yield increases with γ but saturates at a very low value. Incoherent interaction with single target electrons gives rise to two additional bremsstrahlung components, a modest component due to scattering of virtual photons of the electrons on the projectile and a strong low-energy component due to scattering of the virtual photons of the projectile on the electrons. The difference in radiation levels can be traced to the mass of the scatterer. Since target electrons are more widely distributed than nuclei in a crystal channel the variation of the electron component of the bremsstrahlung with incidence angle to a major crystallographic direction is less abrupt than the variation of the nuclear component. In consequence, the shape of the total bremsstrahlung spectrum changes when the crystal is tilted and the individual components may be singled out. Pair creation is also sensitive to the orientation of a crystalline material, resulting in a pronounced directional dependence of the energy loss of bare heavy ions at extreme relativistic energies.