Tea tree oil cuts through this kind of crud with no trouble at all

Just daub a bit onto a cloth and wipe. Follow up with a quick scrub using your favourite old toothbrush and some warm water + detergent. Dry throughly before re-assembling.
Seriously, just last night I used tea tree oil to clean a set of disgustingly filthy buttons from an old cab. At first I thought I had a terrible job ahead of me, but it turned out to be no problem at all with the tea-tree oil.
A bottle of tea-tree oil costs about $2 at the supermarket in Australia, maybe $4 elsewhere I guess. Also makes a fantastic disinfectant and mosquito repellent.