ESR measurements at the X band and the Q band were made on a quasi-one-dimensional planar ferromagnet (CH3)4NNiBr3 (TMNB) in both paramagnetic and three-dimensionally ordered phases. A new resonance associated with the spin motions in ferromagnetic chains was found. This resonance is characterized by an isotropic resonance field and a maximum intensity at 5 K. The resonance fields of the other observed signals show a large out-of-plane anisotropy as well as a distinct sixfold in-plane anisotropy with the symmetry axes deviated from the hexagonal axes by 18°. The signals were assigned to the resonance modes in the spin-floplike phase and the ferromagnetic one. These signals were found to be truncated by the magnetic phase transition in the external magnetic field accompanying a prominent hysteresis. The resonance signal of the spin-floplike phase was found to survive even above the three-dimensional antiferromagnetic transition temperature TN. These results are discussed on the basis of a phenomenological model.