Weighted labelled transition systems are LTSs whose transitions are given weights drawn from a commutative monoid, subsuming a wide range of systems with quantitative aspects. In this paper we extend this theory towards other behavioural equivalences, by considering semirings of weights. Taking advantage of this extra structure, we consider a general notion of weak weighted bisimulation, which coincides with the usual weak bisimulations in the cases of non-deterministic and fully-probabilistic systems. We present a general algorithm for deciding weak weighted bisimulation. The procedure relies on certain systems of linear equations over the semiring of weights hence it can be readily instantiated to a wide range of models. We prove that these systems admit a unique solution for Ω-continuous semirings.