Based on coupled nonlinear Schrdinger equations (CNLSEs) and the split-step Fourier method (SSFM), the ultrashort optical pulse propagation and switching in active three-core nonlinear fiber couplers are numerically investigated. The effects of linear gain coefficient and finite gain bandwidth on pulse propagation and switching have been highlighted, considering the influences of the first and second order coupling coefficient dispersion. It is demonstrated that the linear gain coefficient can sharpen the switching characteristic, lower considerably the switching critical energy, and increase significantly the switching efficiency. Though the finite gain bandwidth of the linear gain degrades the switching characteristics, it not only can suppress significantly the pulse compression and amplification induced by linear gain, but also can suppress effectively the pulse distortion even pulse breakup due to the first order coupling coefficient dispersion as well as the pulse high-frequency fluctuation caused by the second order coupling coefficient dispersion. Because of the effects of the finite gain bandwidth of the linear gain, optical pulses tend to restore periodical coupling propagation in active three-core nonlinear fiber couplers as in the case of the passive fiber couplers.