Solving linear system with sum of kronecker products

Sorry, I still only have a vague clue about preconditioning despite reading up about it in the context of some methods (particularly CG).

Is there a more robust method that is “matrix-free”, in the sense the evaluating Ax is sufficient?

Can you please explain or provide a link (does a package do this? Kronecker.jl seems to provide a type like this but no \ method as far as I can tell).