Convection-driven dynamos in rotating spherical fluid shells are analyzed numerically. A Galerkin method is used for the expansion of the dependent variables in spherical harmonics and in complete systems of functions of the radial coordinate satisfying the respective boundary conditions at the inner and outer spherical boundaries. Forward integration in time is done with a Crank-Nicolson scheme. It appears that time-dependent chaotic solutions tend to exist at lower magnetic Reynolds numbers than stationary dynamos. The magnetic field typically exhibits an oscillatory component with maximum strength at low latitudes. The oscillation period is related to the circulation time in the convection rolls which are nearly stationary in contrast to the motion in the non-magnetic case at the same parameter values. The axisymmetric components of the magnetic field are relatively small and exhibit aperiodic variations on a long time-scale. Although applications to the problem of the origin of geomagnetism have not been the primary goal of the analysis, some tentative conclusions about the geodynamo can be drawn.