We consider two generalizations of series-parallel posets and investigate their complexity with respect to some well-known combinatorial optimization problems which are polynomially solvable on series-parallel posets. Among them are: the jump number problem, the isomorphism problem and the 1|prec|∑ w j C j scheduling problem (minimizing the sum of weighted completion times on one machine). The first generalization is the class of N-free posets, for which the jump number problem is known to be polynomially solvable. But we show that both other problems are, however, hard on this class (i.e., isomorphism-complete and NP-complete, respectively). On the other hand, all three problems and some other such as counting the number of linear extensions or determining the dimension turn (or have turned) out to be polynomially solvable for the second generalization, which consists of all posets obtained by substitution (lexicographic sum) from indecomposable posets of fixed size (posets of bounded diameter). This adds some evidence to our opinion that, with regard to computational complexity, the second class constitutes a much more appropriate generalization of series-parallel posets.