We measured velocities of 862 individual red giant stars in seven isolated dwarf galaxies in the Local Group: NGC 6822, IC 1613, VV 124 (UGC 4879), the Pegasus dwarf irregular galaxy (DDO 216), Leo A, Cetus, and Aquarius (DDO 210). We also computed velocity dispersions, taking into account the measurement uncertainties on individual stars. None of the isolated galaxies is denser than the densest Local Group satellite galaxy. Furthermore, the isolated dwarf galaxies have no obvious distinction in the velocity dispersion-half-light radius plane from the satellite galaxies of the Milky Way and M31. The similarity of the isolated and satellite galaxies' dynamics and structural parameters imposes limitations on environmental solutions to the too-big-to-fail problem, wherein there are fewer dense dwarf satellite galaxies than would be expected from cold dark matter simulations. This data set also has many other applications for dwarf galaxy evolution, including the transformation of dwarf irregular into dwarf spheroidal galaxies. We intend to explore these issues in future work.