  The currently available results of the isotope effect on the superconducting
transition temperature T_c in Fe-based high-temperature superconductors (HTS)
are highly controversial. The values of the Fe isotope effect (Fe-IE) exponent
\alpha_Fe for various families of Fe-based HTS were found to be as well
positive, as negative, or even be exceedingly larger than the BCS value
\alpha_BCS=0.5. Here we demonstrate that the Fe isotope substitution causes
small structural modifications which, in turn, affect T_c. Upon correcting the
isotope effect exponent for these structural effects, an almost unique value of
\alpha~0.35-0.4 is observed for at least three different families of Fe-based
HTS.
