Nonlinear adiabatic dynamics of electron-phonon models of /ital trans/-(CH)/sub /ital x//, /ital cis/-(CH)/sub /ital x//, and (/ital AB/)/sub /ital x//-type polymers are studied. Using both analytic low-amplitude expansion and numerical simulations, we find the existence of charge-neutral breathers---spatially localized, time-periodic nonlinear excitations. Simulations also show that a photoexcited electron-hole pair generates a breather in addition to a soliton-antisoliton pair. Semiclassical quantization shows that the /ital n/th breather is a bound state of /ital n/ amplitude phonons. A semiclassical formalism of optical absorption is applied to /ital trans/-(CH)/sub /ital x//; the ground-state result accounts for a pronounced intragap absorption tail, while breathers can account for the observed photoinduced absorption at 1.35 eV.