Depends what you mean by 'dirty'?

If you mean 'polluted', then it is not very polluted anymore, in fact it is the cleanest it has been in 150 years. Many species of fish have recolonized it. Dolphins and seals are seen swimming in central London.

http://www.telegraph.co.uk/news/earth/wildlife/8059970/The-clean-up-of-the-River-Thames.html

If you mean 'dirty' as in brown and carrying dirt, then the Thames does carry a load of silt, and the effect of tides in its lower reaches (through London and towards the sea) means that the silt on the river bed is constantly being stirred up and suspended in the water.