A combination of three superheated drop detectors with different neutron energy responses was developed to evaluate dose-equivalent and energy distributions of photoneutrons in a phantom irradiated by radiotherapy high-energy x-ray beams. One of the three detectors measures the total neutron dose equivalent and the other two measure the contributions from fast neutrons above 1 and 5.5 MeV, respectively. In order to test the new method, the neutron field produced by the 10 cm X 10 cm x-ray beam of an 18 MV radiotherapy accelerator was studied. Measurements were performed inside a tissue-equivalent liquid phantom, at depths of 1, 5, 10 and 15 cm and at lateral distances of 0, 10, and 20 cm from the central axis. These data were used to calculate the average integral dose to the radiotherapy patient from direct neutrons as well as from neutrons transmitted through the accelerator head. The characteristics of the dosimeters were confirmed by results in excellent agreement with those of prior studies. Track etch detectors were also used and provided an independent verification of the validity of this new technique. Within the primary beam, we measured a neutron entrance dose equivalent of 4.5 mSv per Gy of photons. It was observed that fast neutrons above 1 MeV deliver most of the total neutron dose along the beam axis. Their relative contribution increases with depth, from about 60% at the entrance to over 90% at a depth of 10 cm. Thus, the average energy increases with depth in the phantom as neutron spectra harden.