The authors obtain some new results about global periodicity of general difference equations of order two and three. To be more precise, they find: all the 4-cycles of the form \(x_{n+2}=f(x_{n+1})g(x_n)\), all the 6-cycles of type \(x_{n+3}=f(x_{n+2})g(x_{n+1})x_n\), and all the 3,4 and 5-cycles of the form \(x_{n+3}=f(x_{n+2}, x_{n+1})(1/x_n)\), where the mentioned continuous maps and the initial conditions are positive and most of the \(p\)-cycles obtained enjoy a potential form. It is also proved that all the \(p\)-cycles of the form \(x_{n+2}=f(x_{n+1})g(x_n)\) must verify \(g(x)=c/x\) for some \(c>0\) constant, and \(p\geq 5\).