  Spectral triples over noncommutative principal $\T^n$-bundles are studied,
extending recent results about the noncommutative geometry of principal
U(1)-bundles. We relate the noncommutative geometry of the total space of the
bundle with the geometry of the base space. Moreover, strong connections are
used to build new Dirac operators. We discuss as a particular case the
commutative case, the noncommutative tori and theta deformed manifolds.
