A novel technique for beta-delayed neutron spectroscopy has been developed using trapped radioactive ions. The neutron energy spectrum was reconstructed by measuring the time of flight (TOF) of the nuclear recoil following neutron emission, thereby avoiding all the challenges associated with neutron detection such as backgrounds from scattered neutrons and gamma rays and complicated detector-response functions. A proof-of-principle measurement was conducted on 137I+ by delivering ions from a 252Cf source, confining them in a linear Paul trap surrounded by radiation detectors, and measuring the neutron energy spectrum and branching ratio by detecting the beta and recoil ions in coincidence. Systematic effects were explored by determining the branching ratio three ways. Improvements to achieve higher detection efficiency, better energy resolution, and a lower neutron energy threshold were implemented by upgrading the detectors and optimizing the trapping apparatus. These improvements were demonstrated in a campaign of measurements including 138I, which is a standard beta-delayed neutron precursor outlined by a recent IAEA report on delayed neutron measurements.