A novel multiscale method for discrete fracture modeling on unstructured grids (MS-DFM) is developed. To this end, the DFM fine-scale discrete system is constructed using unstructured conforming cells for the matrix with lower-dimensional fracture elements placed at their interfaces. On this unstructured fine grid, MS-DFM imposes independent unstructured coarse grids for the fracture and matrix domains. While the conservative coarse-scale system is solved over these coarse-grid cells, overlapping dual-coarse blocks are also formed in order to provide local supports for the multiscale basis functions. To increase the accuracy, but maintaining the computational efficiency, fracture-matrix coupling is considered only for the basis functions inside the matrix domain. This results in additional (enriching) fracture basis functions in the matrix. By construction, basis functions form the partition of unity for both fracture and matrix sub-domains. Furthermore, to enable error reduction to any desired level, a convergent iterative strategy is developed, where MS-DFM is employed along with a fine-scale smoother in order to resolve low- and high-frequency modes in the error. The performance of MS-DFM is assessed for several 2D and 3D test cases. Specially, the method proves accurate for many test cases without any iterations. MS-DFM is the first of its kind, and extends the application of multiscale methods to unstructured discrete fracture models. As such, it provides a promising framework for real-field application of unstructured DFM.