I use Barkeeper's Friend too. I find other cleansers, like Comet, work just as well though. Barkeeper's Friend has more machismo in the name. I don't know if the grits are different but the result seems the same.
They all work better on Scotch Brite then on cloth. I have not tried them with a tooth brush but that sounds tedious as it only takes a few full length strokes with the cleanser on a little piece of Scotch Brite pad if done properly (see below). I keep a piece of Scotch Brite pad about 3" or 4" square in my sharpening kit. The green or maroon commercial ones both seem to work equally well as does the green pot scrubbing Scotch Brite stuff on the back of some sponges. I don't think the Scotch Brite is doing the work as much as it just holds the cleanser better then any thing else. A sponge works almost as well ir it is not too wet and with a lot of cleanser but tends to wash away the cleanser if it is too wet or you squeeze too hard. Scotch Brite doesn't hold much water.
It works better (faster) when you use a lot of cleanser with some depth to it. I wet the Scotch Brite and dump a liberal amount of cleanser, about 1/4" deep, onto the pad and add a few more drops of water (not enough to wash the cleanser away or reduce its depth). When you start scrubbing the mass should be fairly dry and then get wet with a few strokes and foam up as you scrub.
I find if I just sprinkle a little cleanser on the Scotch Brite, like I would for normal cleaning tasks with a cloth, it works, but takes quite a bit of scrubbing to get all the metal out. When applied heavily, like this, it only takes a few strokes and the metal is magically all gone. If I get a section where the metal didn't all come out and the cleanser has started to thin out it starts to take a lot of scrubbing again. Dump a bunch of cleanser on the pad again and the remaining metal comes right out immediately.
The clean rod is much more aggressive then when it has some metal in it (is black) and cleaning should be done often to keep your ceramic rods working like they should.