The sensitivity of the turbulent flame speed to the geometry of the flame is investigated using direct numerical simulations of turbulent premixed flames in three canonical configurations: freely propagating statistically planar flames, planar flames stabilized in stagnating flows, and rod-stabilized V-flames. We consider both the consumption speed, which measures the integrated rate of burning, and the propagation speed, which measures the speed of an isosurface within the flame brush. An algebraic model for the propagation speed of the leading edge of the flame brush, which is blind to flame geometry, is also applied to the data for the purposes of establishing its range of validity and the causes of its failure. The turbulent consumption speed is found to be strongly geometry dependent, primarily due to the continuous growth of the flame brush thickness. Changes in the structure and consumption speed of instantaneous flame fronts are found to be only weakly sensitive to flame geometry. The turbulent propagation speed is analysed in terms of its reactive, diffusive and turbulent flux components. All three terms are shown to be significant, both through the flame brush and along the leading edge. The leading-edge propagation speed is found to be sensitive to flame geometry only in the V-flames under certain conditions. It is suggested that this apparent geometry dependence, which the model cannot capture, results from the relation between the turbulence and mean flow time scales in these particular cases, and is not intrinsic to the flame geometry itself.