Elliptic interface problems with multi-domains and triple junction points have wide applications in engineering and science. However, the corner singularity makes it a challenging problem for most existing methods. An accurate and efficient method is desired. In this paper, an efficient non-traditional finite element method with non-body-fitting grids is proposed to solve the elliptic interface problems with multi-domains and triple junctions. The resulting linear system of equations is positive definite if the matrix coefficients for the elliptic equations in the domains are positive definite. Numerical experiments show that this method is about second order accurate in the $L^∞$ norm for piecewise smooth solutions. Corner singularity can be handled in a way such that the accuracy does not degenerate. The triple junction is carefully resolved and it does not need to be placed on the grid, giving our method the potential to treat moving interface problems without regenerating mesh.