See all those rolled up balls of dust? That is what happens when you try and wipe away the oily dust with a dry rag.
What is the solution? Fight oil with oil.
It will take a lot of soap and water (or any other type of cleaner) to get rid of oil splatters or oily dust. Instead, put a couple drops of mineral oil on a paper towel. Wipe the paper towel over the dirty surface, and watch the gunk come right off.
When you are done cleaning, simply wipe down the surfaces one more time with a new paper towel. This leaves the surface coated lightly in oil, which makes future oil splatters much, much easier to clean up.
Updated 5/23: Although vegetable oil works fine, a chemist reader of mine explains why mineral oil is the better long-term cleaning choice.
“Short-term, vegetable oil does work just as well, for the shared common property of ‘oiliness’ that you’re taking advantage of here. Longer-term, mineral oil has the benefit, over vegetable oil, that it will not ‘yellow’ and will not get tacky or gummy, as will vegetable oil. The reason that vegetable oil is good for your diet – polyunsaturated fatty acids – is also the reason that it goes yellow and gets sticky with prolonged exposure to oxygen in the air.”