If you mean the outside of the bowls, I use a microfiber cloth, with some saliva if the bowl appears to be dirty. I would usually never put alcohol on the outside of a pipe, but since this one is unstained I could see a potential benefit to everclear. Since the untreated wood is absorbent perhaps something that will evaporate off would be a good idea. Not sure about that. I have used coffee on a rag on rim char with good results.
If you mean the inside of the bowl, I usually just ream em instead of wiping it out.
Hmm, yeah, the exterior on a sav unfinished is kind of a pain if you're looking to keep it clean as it collects any dirt and oils from your hands, whatever it happens to get on it, etc.. Like Stompy suggested, a microfiber towel and water or a dab of alcohol is probably your best bet since you really don't want to use cleaning chemicals on the pipe as it will absorb into the briar, which could range from poisonous to just tasting nasty. If it were really messed up I could see very fine sandpaper being used, but that's a whole nother can of worms.
Gnossos said that he makes a pot of strong black coffee, lets it cool, then dips a rag in it & uses that to get rid of all the dirt and oil build up on his natural vergin castellos. I personally have never tried it since I like em all grubby. He said the coffee itself doesn't leave any stains. Good luck!
The coffee trick is great, cleans well and will break down oils to a certain extent, and surprisingly (especially if you've ever spilled any on a white shirt) won't stain the wood. That being said, I don't think it's possible to get unstained briar back to the light color of when it was new. Even if you held it by the stem all the time, the briar would still color just from being smoked (which is what DGE is talking about 2 posts back.)
A natural unstained briar pipe bowl will darken/patina over time. It was already stated that this is normal, and I can tell you that you will never get the bowl to stay/remain a light color. I have a BST bent that SAS made that was like ths. It now has an absolutely stunning mahogany patina to the bowl that is beautiful. I have seen this process in countless pipes. It's normal, and if one wants that virgin unstained look to remain--well I have never seen a way to make this possible. It's the natural process of heat, and such like Doug said--picture a red uncooked steak--as it heats it turns color. I cannot ecplain the science behind what happens with a briar pipe bowl, but I know what happens to the color, and it cannot be prevented. On my BST that I currently cannot post pictures of the bowl color is actually beautiful. It's a nce mahogany color that no stain could ever emulate so well. I love how my BST has colored.
Now granted some may not like this to happen, but I have to see a way to stop it from happening.
Outside of the bowl I rub either side of my nose and the natural oils of the skin add a shine to the pipe then I rub with a microfiber cloth.
Inside once a month or so I'll dip a pipe cleaner in some whiskey, doesn't have to expensive stuff a shops own brand will do however it must be drinkable to you, run it around the inside of the bowl and leave it to fully dry.
The reason why I say it has to be drinkable to younis if it is a rocket fuel gut rot type of whiskey or spirit is it can leave a taste in the pipe but a drinkable whiekeynwill sweeten the pipe and get it back to its smoking best.