  We study spectral triples over noncommutative principal U(1)-bundles of
arbitrary dimension and formulate a compatibility condition between the
connection and the Dirac operator on the total space and on the base space of
the bundle. Examples of low dimensional noncommutative tori are analyzed in
more detail and all connections found that are compatible with an admissible
Dirac operator. Conversely, a family of new Dirac operators on the
noncommutative tori, which arise from the base-space Dirac operator and a
suitable connection is exhibited. These examples are extended to the
theta-deformed principal U(1)-bundle S^3_\theta -> S^2.
