Right okay, we've got a washing machine.

The problem is - it's making my clothes dirty.

Basically, clothes go in smelling like they need a clean and come out smelling of damp/mould.

I've been leaving the washing machine door open at all times, but that doesn't seem to have helped much.

Anyone got a solution?

Have you taken the rubber seal out from the door and cleaned it? Have you cleaned the inside of the washing machine?

Have you checked the drawer where the washing powder goes? It's disgusting but put your hands down the back of that drawer and check for any gunk and pull it out, sometimes detergent builds up here and turns into scum (especially powder). Try to put your detergent in the drum in future instead of the drawer if this is the problem.

I've cleaned the rubber seal, but haven't taken the drawer out - I'll give that a try.

I also tried putting a wash on with no detergent or clothes, to see if that would get rid of it but it hasn't.

My housemate wants to put rice and bicarb in there overnight, thinking the rice will take out all the moisture and bicard will take out the odor.

Sounds like voodoo to me!!

As well as the other advice, try putting some bleach through it on an empty load, on a boil setting (or as hot as your machine will go.)
Might also be worth thinking about putting some of those tablets to dissolve any calcium build up, as I'd imagine any limescale will hold the bacteria that make it smell.
But probably not at the same time as the bleach!

Cleaning the washing powder tray and the "collector" under the flap at the bottom are good places to start. The fluff collector remains wet and stinks to high heaven if you don't clean it out on a regular basis (every 10 loads or so). (Some machines don't have a fluff collector though.)

When mine started to stink, I did like Alex and ran it empty with bleach. The other is to run it on a 90° cycle, that should kill off the bacteria causing the smell - and more environmentally friendly than pumping bleach into the sewers (one of my girlfriend's brothers did some sort of degree on environmental protection).

As already mentioned, run the machine on a boil wash with either a small amount of bleach or some bleach-containing powder. If you use bleach, dilute it and pour it down all compartments in the detergent drawer to make sure the feed pipes are treated (just don't pull the drawer out too far or it'll end up down the front of the machine). Also, remove the filter and make sure it's clear (ou can soak it in dilute bleach as well if necessary).

The tablets for shifting limescale aren't much use in this case as they only work if you use them repeatedly.

The internal pipes might need cleaning. You can buy chemicals that clear away and dissolve any crap in there.
FOR EXAMPLE

Don't bother - they're not worth the money. The tablets are only made of bleach, sodium carbonate and boric acid so don't possess any "magical" cleaning properties - buy a tub of Vanish and it'll have much the same things in it.