Using recently proposed method of discrete Hirota dynamics for integrable (1 + 1)D quantum field theories on a finite space circle of length L we derive and test numerically a finite system of nonlinear integral equations for the exact spectrum of energies of SU(N) x SU(N) principal chiral field model as functions of mL, where m is the mass scale. We propose a determinant solution of the underlying Y-system, or Hirota equation, in terms of Wronskian determinants of N x N matrices parameterized by N - 1 functions of the spectral parameter theta with the known analytic properties at finite L. Although the method works in principle for any state, the explicit equations are written for states in the U(1) sector only. For N > 2, we encounter and clarify a few subtleties in these equations related to the presence of bound states, absent in the previously considered N = 2 case. As a demonstration of efficiency of our method, we solve these equations numerically for a few low-lying states at N = 3 in a wide range of mL. (C) 2015 The Authors. Published by Elsevier B.V.