A velocity imaging technique combined with (2+1) resonance-enhanced multiphoton ionization (REMPI) is used to detect the primary Br((2)P(3/2)) fragment in the photodissociation of o-, m-, and p-dibromobenzene at 266 nm. The obtained translational energy distributions suggest that the Br fragments are produced via two dissociation channels. For o- and m-dibromobenzene, the slow channel that yields an anisotropy parameter close to zero is proposed to stem from excitation of the lowest excited singlet (pi,pi*) state followed by predissociation along a repulsive triplet (n,sigma*) state localized on the C-Br bond. The fast channel that gives rise to an anisotropy parameter of 0.53-0.73 is attributed to a bound triplet state with smaller dissociation barrier. For p-dibromobenzene, the dissociation rates are reversed, because the barrier for the bound triplet state becomes higher than the singlet-triplet crossing energy. The fractions of translational energy release are determined to be 6-8 and 29-40 % for the slow and fast channels, respectively; the quantum yields are 0.2 and 0.8, and are insensitive to the position of the substituent. The Br fragmentation from bromobenzene and bromofluorobenzenes at the same photolyzing wavelength is also compared to understand the effect of the number of halogen atoms on the phenyl ring.