The Birchwood-Casey Lead - away Wipe Rag works well - but so does a pencil eraser. Just make sure to not get the debris between parts.
Unless you plan on not firing the gun for a good amount of time, I'd not get crazy about the "rings" around the Cylinder Face. As soon as you fire the first cylinder full of ammo it will be like you never cleaned them! I use solvent and a soft nylon "toothbrush" to clean the face of the Cylinder when cleaning. All that really does is it removes the heavy loose stuff but does not take off the rings. This prevents a thick build-up and will allow the Cylinder to remain free of jamming against the Forcing Cone for good operation.
If you are turning a Revolver into a Safe Queen and want it in pristine shape, then use the lead Wipe mentioned above and by others or a pencil eraser. If it's a regular shooter, just keep it as clean as a toothbrush and solvent will get it.
Just my opinion and YMMV.