Wet a terrycloth with warm water and rub it in the direction of the grain over the stain. Do not rub in circles or against the grain as this will push the stain into the material and make it harder to remove. If warm water doesn't work go to the next step.
Dip the terrycloth in vinegar and rub it over the stain in the same manner as with the water. This can result in the area you are cleaning becoming shiny as new (and you might have to clean the entire piece to make it look that way). For utensils you can try removing stubborn stains by soaking them in vinegar overnight.
Create a slurry of vinegar and baking soda. Rub this on any tough stains with your index finger or a terrycloth. The grit in the baking soda will pull deeper stains out without scratching the surface of the steel.