In order to obtain the sequences with a few autocorrelation values and large linear complexity, a new class of binary cyclotomic sequences of order 3 with period p were constructed, where p is a prime and p≡1(mod 3). The autocorrelation was computed based on cyclotomy, and the condition for p that assures the 3-valued autocorrelation was discussed. The condition is that p should be the form p= a ²+12 for an integer a. The linear complexity is p-1 if p is the form, or 2( p-1)/3 otherwise. By computer experiments, all ps' satisfying the form were found, the corresponding sequences were given, and the autocorrelation and linear complexity were confirmed. The linear complexity was the same as that of the known ternary cyclotomic sequence of order 3. Compared with the related known binary cyclotomic sequences of even order, the linear complexity was the same or better in most cases. The method in this paper can be extended to construct other cyclotomic sequences of odd order with a few autocorrelation values and large linear complexity. Since the cyclotomic sequences of larger odd order also have better balance, they can be applied to stream ciphers and communication systems.