The rate coefficients alpha for the dissociative recombination (DR) of the aromatic cyclic ions C6H6+ and C10H8+ have been measured in a flowing afterglow experiment at room temperature using Xe+ and Kr+ as precursor ions. C+ was also used as a precursor ion and this allowed the DR rate coefficients for C3H3+, C5H3+, and C7H5+ to be determined. The values obtained (in units of 10(-6) cm3 s-1) at 300 K are, respectively, alpha(C3H3+) = 0.7, alpha(C5H3+) = 0.9, alpha(C6H6+) = 1.0, alpha(C7H5+) = 0.7, alpha(C10H8+) = 0.3 with an uncertainty estimated to be approximately 30%. A discussion is given concerning the relationship between the DR rate and the structural complexity of the ions.