Consider the 6th-order triharmonic problem in a polygonal domain with the simply supported boundary condition. The finite element approximation to such high-order problems often involves sophisticated construction of the bilinear form and of the finite element space. We propose a new C0 finite element algorithm to solve these problems. Our methods are intuitive, easy to implement, and applicable to both convex and nonconvex domains. We give finite element error analysis and report numerical results to validate the algorithms.