A new and efficient computational approach is presented for analyzing transient heat conduction problems. The Fragile Points Method (FPM) utilized for spatial discretization can, on one hand, like many other meshless methods, be free of the requirement of high-quality meshing, and on the other hand, bypass the difficulty of domain integration problem which is commonly seen in Galerkin meshfree methods. With local, polynomial and discontinuous trial and test functions, the method has a great potential in solving problems with rupture and fragmentation without remeshing. Anisotropy and nonhomogeneity which is challengeable for many spatial numerical methods do not give rise to any difficulties in the present implementation. The Local Variational Iteration Method (LVIM) in the time domain is a highly efficient technology in solving nonlinear problems, in which the time steps can be an order of magnitude larger than the traditional backward Euler scheme and the computing time can be cut by a half. The FPM+LVIM solver is also connected to the prepossessing module of ABAQUS which helps generating the domain partition. It shows the compatibility of the current approach with various partitions and makes it more friendly for engineer users. Several 2D and 3D numerical examples with functionally graded and composite materials are then provided as validations.