We present single-dish H I spectra obtained with the Green Bank Telescope, along with optical photometric properties from the Sloan Digital Sky Survey, of 20 nearby (D < 70 Mpc) Luminous Compact Blue Galaxies (LCBGs). These ~L*, blue, high surface brightness, starbursting galaxies were selected with the same criteria used to define LCBGs at higher redshifts. We find these galaxies are gas-rich, with M(HI) ranging from 5*10^8 to 8*10^9 M_sun, and M(HI)/L_B ranging from 0.2 to 2 M_sun/L_sun, consistent with a variety of morphological types of galaxies. We find the dynamical masses (measured within R_25) span a wide range, from 3*10^9 to 1*10^11 M_sun. However, at least half have dynamical mass-to-light ratios smaller than nearby galaxies of all Hubble types, as found for LCBGs at intermediate redshifts. By comparing line widths and effective radii with local galaxy populations, we find that LCBGs are consistent with the dynamical mass properties of Magellanic (low luminosity) spirals, and the more massive irregulars and dwarf ellipticals, such as NGC 205.