  We study some of the novel effects that arise when the QCD axion is placed in
the ``bulk'' of large extra spacetime dimensions. First, we find that the mass
of the axion can become independent of the energy scale associated with the
breaking of the Peccei-Quinn symmetry. This implies that the mass of the axion
can be adjusted independently of its couplings to ordinary matter, thereby
providing a new method of rendering the axion invisible. Second, we discuss the
new phenomenon of laboratory axion oscillations (analogous to neutrino
oscillations), and show that these oscillations cause laboratory axions to
``decohere'' extremely rapidly as a result of Kaluza-Klein mixing. This
decoherence may also be a contributing factor to axion invisibility. Third, we
discuss the role of Kaluza-Klein axions in axion-mediated processes and decays,
and propose several experimental tests of the higher-dimensional nature of the
axion. Finally, we show that under certain circumstances, the presence of an
infinite tower of Kaluza-Klein axion modes can significantly accelerate the
dissipation of the energy associated with cosmological relic axion
oscillations, thereby enabling the Peccei-Quinn symmetry-breaking scale to
exceed the usual four-dimensional relic oscillation bounds. Together, these
ideas therefore provide new ways of obtaining an ``invisible'' axion within the
context of higher-dimensional theories with large-radius compactifications.
