We demonstrate chirped-pulse operation of a Cr : YAG passively mode-locked laser. Different operation regimes of the laser are extensively investigated in the vicinity of zero dispersion both experimentally and numerically. It is shown that for a given laser configuration, transition to the positive dispersion regime allows a 5-fold increase in the output pulse energy, which is otherwise limited by the onset of the multipulsing or 'chaotic' mode-locking. The output pulses have 1.4 ps duration and are compressible down to 120 fs in a 3 m piece of silica fiber, enabling supercontinuum generation in a nonlinear fiber. The spectrum shape and operation stability of the chirped-pulse regime depend strongly on the amount and shape of the intracavity dispersion. The numerical model predicts the existence of the minimum amount of the positive dispersion, above which the chirped-pulse regime can be realized. Once located, the chirped-pulse regime can be reliably reproduced and is sufficiently stable for applications.