We introduce block versions of the multi-elimination incomplete LU (ILUM) factorization preconditioning technique for solving general sparse unstructured linear systems. These preconditioners have a multi-level structure and can be shown to have some properties that are typically enjoyed by multigrid methods. Several heuristic strategies for forming blocks of independent set are introduced and their relative merits are discussed. Advantages of block ILUM over point ILUM include increased robustness and efficiency. We compare several versions of the block ILUM, point ILUM and the dual-threshold-based ILUT preconditioners. In particular, the ILUM preconditioned Krylov subspace solver is tested for some convection-diffusion problems to show convergence that is near Reynolds number independent and near grid independent.