London is as safe or dangerous as you make it.
I don't think I'm ever at full ease but to sum up the dangers, on the same day the 7/7 bombs went off, one of my friends was stabbed and killed by someone he didn't know or speak to in Romford, which is in zone 6 on the tube and apparently a safe area. Well it is safe for the most of it.
Just be mindful, keep your belongings safe, like wallet in a chest or front jeans pocket rather than back jeans pocket or in a bag. If you have a laptop, use a rucksack and put the laptop into a sleeve inside it. Also get non standard headphones for your iPod/iPhone.
Also, my dad is a Police Officer and his main jurisdiction is on the Underground, there have been reports of pickpockets dip your wallet, they have a smartcard reader, read the card, put it back in the wallet and put the wallet back in your jeans/bag.