High energy X-ray beams from medical accelerators are used in cancer therapy. In such beams, neutrons are also produced due to photon-neutron interactions in the target material, collimator and beam flattening filter. The feasibility of employing fast-neutron-induced recoil particle tracks in polycarbonate foils developed by the electrochemical etching (ECE) method for the dosimetry of such neutrons was investigated. The experiments were carried out in, and out of, the beam of a Varian Clinac-18, an Allis-Chalmers 25 MeV betatron and a Brown Boveri 45 MeV betatron. Neutron dose equivalents were found both as a function of the photon dose and of the distance away from the centre of the beam. The neutron dose equivalent was found to be 0.6 and 1.3% of the X-ray dose at the centre position of the beam for a TSD = 100 cm and a 10 x 10 cm2 field for the 25 and 45 MeV betatrons respectively. Although the dosemeter showed adequate sensitivity to neutrons in the beam of the betatrons, it showed insensitivity to neutrons in the beam of the Clinac-18 which has a softer spectrum than the neutron energy threshold of the dosemeter. It can be concluded that this dosimetry method provides a new approach with a number of advantages compared to some other existing techniques for such measurements above the energy threshold of the dosemeter.