An algorithm that finds a set of real-valued approximately-zero correlation zone (AZCZ) sequences is proposed on the basis of the concept of feedback-controlled direct-sequence code-division multiple access (FC/DS-CDMA). It is known that ordinary algorithms can construct low correlation zone (LCZ) and zero correlation zone (ZCZ) sequence sets in which the choices of the number of sequences, sequence length, and LCZ or ZCZ length are limited. It is shown that the proposed algorithm finds AZCZ sequence sets by a numerical method under arbitrary conditions. The properties of AZCZ sequence sets are evaluated in terms of the autocorrelation and cross-correlation functions. It is shown that the periodic autocorrelation and cross-correlation functions take small values within a designated AZCZ. It is also shown that we can construct approximately-perfect sequences that have approximately ideal autocorrelation functions and new sequence sets that have multiple AZCZs using the proposed algorithm.