Quantum group Fourier transform methods are applied to the study of processes on noncommutative Minkowski space-time [xi, t]=ιλxi. A natural wave equation is derived and the associated phenomena of in vacuo dispersion are discussed. Assuming the deformation scale λ is of the order of the Planck length one finds that the dispersion effects are large enough to be tested in experimental investigations of astrophysical phenomena such as gamma-ray bursts. We also outline a new approach to the construction of field theories on the noncommutative space-time, with the noncommutativity equivalent under Fourier transform to non-Abelianness of the ``addition law'' for momentum in Feynman diagrams. We argue that CPT violation effects of the type testable using the sensitive neutral-kaon system are to be expected in such a theory.