Sorry for the Rez-post, but if I didn't know about this, I'd want to:
Tthere is a non-toxic solvent option made from orange peel essential oil, and it dissolves grease rapidly and effectively as any petroleum product. It's called Citra-Solv, and is sold widely (they sell it at my local grocery stores here in the SF Bay Area). It's not proprietary (It's just Limonene extracted from Orange peel) but it's the only brand I know of here. I use it for literally everything where stripping old oil/grease is required (door hinges, removing label adhesive, preparing a keyhole for lubricant).
As for technique for using it on a chain, this method is simple and works well for me:
[Materials needed: A brush (old toothbrush is OK, but there are better chain-specific products which "cup" the chain), dirty rag, paper towels, oil-proof glove(s)]
-Running along the chain, scrub the chain with the brush, dunking the brush in the orange oil (in a paper cup or small glass jar) to replenish as needed.
-Using the rag in the off-hand with a paper towel on top to both prevent the orange oil spillling, and wipe the chain. Refresh the paper towels as needed.
(Wearing a latex glove, or vinyl, nitrile, or dish glove, on the off-hand.)
Also, I often use orange oil and paper towels to clean a frame up in a jiffy (old chain and axel grease somehow ends up on there and attracts dust and road schmutz). It's faster and safer than water, and leaves it shiny and clean.
One word of warning though: It is a thin solvent with no water tension, so it will seep rapidly, so bare that in mind. It's easy to apply more than is needed, unless you use a bottle with a dropper to apply it (I often use an old essential oil bottle to apply it when I only need a few drops). It can also be diluted and mixed with water (turns the water milky/opaque when you shake it up), and applied with a spray bottle, for cleaning etc.
And as for a natural non-toxic lubricant, I use Jojoba Oil. It's a liquid wax so it never cross-polymerizes the way natural oils do. It's a bit thin for chains by most people's standards, but I don't mind applying more frequently in order to avoid wasting energy on a sticky chain. (Thick chain lube is a little-known cause of efficiency loss - also harder to remove when it's contaminated). But even if you prefer thicker chain lube, or need to use it for climactic reasons (doesn't rain much here), Jojoba is good for bearings, hinges, keyholes, and anywhere you want a joint to move as smoothly as possible.
