In this paper we present a construction method for quaternary sequences from a binary sequence of even period, which preserves the period and autocorrelation of the given binary sequence. By applying the method to the binary sequences with three-valued autocorrelation, we construct new quaternary sequences with three-valued autocorrelation, which are balanced or almost balanced. In particular, we construct new balanced quaternary sequences whose autocorrelations are three-valued and have out-of-phase magnitude 2, when their periods are N = p^m - 1 and N ≡ 2 (mod 4) for any odd prime p and any odd integer m. Their out-of-phase autocorrelation magnitude is the known optimal value for N ≠ 2, 4, 8, and 16.