i dont know if my way actually gets it clean, but it sure works for my 5 year old linen...and it gets dirty:
take a brown paper bag and cut it into strips about 4-5 inches wide and 6-8 inches long. use the strips and burnish the wrap with the paper: you have to have a pretty firm grip with the paper--make sure you are rubbing with the linen wrap...as in dont stroke it up and down against the wrap, but with the wrap--like its spinning on a lathe.
do this with a couple strips of the paper until the paper is no longer dirty.
now i take kitchen wax paper and cut it into the same size as the paper bags...and then do the same thing. i like to do a couple applications, with a really tight grip and then go over and burnish the wrap with a couple more brown paper bag strips. this will rub the wax in good and remove the excess.
this is actually very hard on your hands...mine end up getting pretty sore after the whole process. i think it might be a tad easier on a lathe, but thats how i do it by hand.