A variety of fractal grids is used to investigate how fractal-grid-generated turbulence affects the turbulent flame speed for premixed flames. The grids are placed inside a rectangular duct and a V-shaped flame is stabilized downstream of the duct, using a metal wire. This flame is characterized using OH-LIF. The turbulent flow is characterized by means of hot-wire anemometry. In total 24 grids containing variation in grid patterns, blockage ratio and range of embedded scales were assessed. The results quantify in detail the relation between grid parameters and the turbulence downstream of the different grids. The increase in turbulent flame speed is found to be primarily related to the level of turbulent fluctuations and the time available for the turbulence to wrinkle the flame. The fractal grids provide efficiently much more intense turbulence compared to classical grids resulting in an increased turbulent flame speed. By increasing the range of embedded scales the turbulence is intensified. With respect to the reference case the turbulence intensity can be more than quadrupled and for the turbulent flame speed a more than doubling is observed. The turbulent flame speed reported here is in agreement with observations by others, while a stronger broadening of the flame front curvature distributions is found. In practice the maximum usable increase in turbulence will be limited by the allowable transverse inhomogeneity in mean velocity that is accompanied with the inhomogeneous blockage of the fractal grid.