Hard carbons prepared from organic precursors by pyrolysis near 1000 degrees C generally have poorly developed (002) Bragg peaks and also show evidence for microporosity in small angle x-ray scattering experiments. When the peak-to-background ratio of the (002) peak is small, it indicates that hard carbons contain significant amounts of single, bilayer, or trilayer graphene sheets which are arranged at arbitrary angles. These carbons also contain the micropores implied by such a stacking arrangement. With electrochemical measurements on Li/carbon electrochemical cells, we show that those carbons with the largest capacity for lithium are those with the largest fraction of single graphene sheets and with the smallest average micropore size. Over 80 samples prepared from a variety of precursors at a selection of temperatures have been studied. The trends observed may help point the way to better anode materials for advanced Li-ion batteries.