Brushed steel in the kitchen, particularly with refrigerators, is very popular, and it looks quite elegant. Unfortunately, it is also a surface that shows residue from human touch and even water very visibly. So if you use a damp cloth to clean it, the residue from the water remains. Very frustrating! There are steel polishes that work well(I recommend the product Wow!) but once you start with one it's hard to stop because the oil or wax from the polish itself becomes very difficult to ever remove entirely.

A simpler solution is this (here's a trade secret!): use a clean damp cloth with water and be sure to wipe it dry vigorously against the grain. That way, the residue from the water has much less contact in the brushed steel's micro-grooves to cling to. 

I'm going through a move right now. Of course it's standard to clean the premises for the next occupant after the furnishings have all been cleared. But it's been interesting to find how much unexpected deep cleaning to my own furnishings a move allows. As I've been getting my things packed and cleared off, I see things I haven't attended to in a long time. Two examples: hazy glass on my framed pictures and the dust that has accumulated behind my books on my bookshelves. It's a nice ritual to get at these accumulations to be off to a clean start at the new place!