For linear dynamic equations x^△ = A(t)x in R^N on a time scale T (e.g. T = Z or T = R) the so-called dichotomy spectrum is introduced in this paper. This new spectrum consists of at most N closed intervals of the real line. In the autonomous case with T = R these intervals reduce to the real parts of the eigenvalues of A. In any case the spectral intervals are associated with invariant vector bundles comprising solutions with a common exponential growth rate. The main result of this paper is a spectral theorem which describes all possible forms of the dichotomy spectrum.