I've lived in many a rented house and although being furnished is a great help, you can never be sure what's "clean" or not. Every time I move into a new place, the first thing I do is give the mattress a good clean with this method and gives me a sweet-smelling bed for the future.
Step 1: What you will need
Cleaning a mattress is very simple and remarkably effective considering the items used. What you will need are:
The mattress that you want cleaned
Some Bicarbonate of Soda (available in all supermarkets in the baking aisle)
A sieve (an icing sugar shaker/duster is even better)
A vacuum cleaner
That is all you need.
Step 2: The Dusting
If you are using a sieve, pour a small amount of bicarbonate of soda into the sieve and use the sieve to sprinkle the soda all over the mattress. You're aiming for a good dusting over the entire surface of the mattress so don't worry too much if you accidentally put too much on, just spread it around.
Areas that have stains or soiling should be treated with more soda on that spot.
If you have a dusting tin (like a big salt/pepper shaker but for icing sugar and cocoa powder), then just fill the shaker up and sprinkle liberally.
Step 3: The Cleanup
After you have dusted the entire surface of the mattress with bicarbonate of soda, leave the mattress for several hours (a whole day if possible) for the soda to work it's magic.
The soda draws dirt, moisture and odours out of the fabric and absorbs them leaving the material unblemished.
After the mattress has been left to settle, use the vacuum cleaner to suck up all of the soda from the surface and inside the mattress. Do this quite thoroughly otherwise you'll get puffs of soda everytime you lie down.