Dielectronic recombination rates have been measured for several charge states of isoelectronic Li-like, Be-like, B-like and Na-like ions. We observed the amount of electron capture attending the passage of MeV/nucleon ion beams through a collinear, magnetically confined space-charge-limited electron beam as a function of relative energy. The experimental rates are consistent in magnitude and shape with rates determined from distorted-wave calculations of the dielectronic-recombination cross sections.