Spectral triples from bimodule connections and Chern connections

We give a geometrical construction of Connes spectral triples or noncommutative Dirac operators $D$ starting with a bimodule connection on the proposed spinor bundle. The theory is applied to the example of $M_2(\Bbb C)$, and also applies to the standard $q$-sphere and the $q$-disk with the right classical limit and all properties holding except for $\mathcal J$ now being a twisted isometry. We also describe a noncommutative Chern construction from holomorphic bundles which in the $q$-sphere case provides the relevant bimodule connection.