D-separation still works for cyclic graphs, it just can't rule out causal relationships between variables that lie on the same cycle. And neither can any other functional-form-agnostic method, because in general feedback loops really do couple everything to everything else.
More rigorously: given a graph G for a structural equation model S, construct a DAG G' as follows
- Find a minimal subgraph C_i transitively closed under cycle membership (so a cycle, all the cycles it intersects, all the cycles they intersect, and so on)
- Replace each C_i with a complete graph C'_i on the same number of vertices, preserving outgoing edges.
- Add edges from the parents of any vertices in C_i (if not in C_i themselves) to all vertices in C'_i
- Repeat until acyclic
d-separation in G' then entails independence in S given reasonable smoothness assumptions I don't remember the details of off the top of my head.
