Recently Ahmadi et al. [IEEE Trans. Parallel Distrib. Syst., 32 (2021), pp. 1452–1464] and Tagliaferro [Research Square (2022)] proposed some iterative methods for the numerical solution of linear systems which, under the classical hypothesis of strict diagonal dominance, typically converge faster than the Jacobi method but slower than the forward/backward Gauss–Seidel one. In this paper we introduce a general class of iterative methods, based on suitable splittings of the matrix that defines the system, which include all of the methods mentioned above and have the same cost per iteration in a sequential computation environment. We also introduce a partial order relation in the set of splittings and, partly theoretically and partly on the basis of a number of examples, we show that such partial order is typically connected to the speed of convergence of the corresponding methods. We pay particular attention to the case of linear systems for which the Jacobi iteration matrix is nonnegative, in which case we give a rigorous proof of the correspondence between the partial order relation and the magnitude of the spectral radius of the iteration matrices. Within the considered general class, some new specific promising methods are proposed as well.