Do you think that there is a strategy in making your linoleum cleaner.

I am not sure if they have it in your country, but pine sol may work better. Or some soft scrub, it is a bleach but it will scrub into the floors as you scrub it and help get some of that ground dirt out. You may want to try a thorough sweep, then vaccuming first, then try to clean the linoleum with the methof I gave you. If that still doesn't help making a nice vinegar scrub after it dries will help. I know vinegar and baking soda help on carpets.