I wouldn't say that fleece cuts down all odours but it does a pretty good job. When I first got guinea pigs I started with newspaper which did not last as it simply is not absorbent. ThenI went to shavings which were ok, and I was able to remove soiled bits and replace with fresh which lengthened the time between cleanings greatly. However it became very expensive. So I then went to towels. Towels by themselves do NOT work, I was changing the cage twice a day! When I added fleece on top, it worked perfectly and I now get a week between cleanings. I would recommend, if you decide to use fleece, to have a litter tray or kitchen area filled with shavings or 100% recycled kitty litter. In this area you put all of their food - so pellets, hay etc. Guinea pigs tend to go to the toilet where they eat, so by having a kitchen area that you can spot clean daily - the space between cleanings is increased.
However, how big is your cage and how many pigs do you have in it? The smaller the cage the more built up their soilings are which lowers the time between cleanings. So not only is a bigger cage better for the pigs - but it makes it easier for you to!
