Measurements of the spin-lattice relaxation time for rhenium in KRe04 and NH4Re04 are reported; relaxation rates are extremely fast due to the large quadrupole moment of Re. Calculations of the electric field gradient in KRe04 show that the EFG is a very sensitive function of the orientation of the anion in the unit cell as well as of the unit cell dimensions, and that geometrical effects make a major contribution to the temperature dependence. Neutron diffraction in these two salts has been employed to determine the temperature dependence of structural parameters over a wide range of temperature, and preliminary results are reported.