According to the this link (in the "Crime" section), I should avoid the areas of "Roppongi, Shinjuku (especially the area of Kabuki-cho), Shibuya, and Ikebukuro" in Tokyo, due to them being high crime areas.
Is there any truth to this?
I'm afraid I can't find any government numbers to back up my anecdotal evidence, but Shinjuku, Shibuya and Ikebukuro are all very safe - much safer than their equivalents in London, New York or any other 'world' city. I've spent many nights in each, in varying states of sobriety, and never had any problems at all. I've lost my wallet a few times (on trains, in bars and the like) and never failed to retrieve it - either at the time from a fellow reveller, or at a police box (koban) later on.
Kabuki-cho is the closest thing you'll get to a 'red light' district. Plenty of 'love hotels', hostess clubs, adult shops and yakuza (Japanese mob) to go around. Even so, much like Amsterdam's equivalent, it's as much an entertainment spot and centre of nightlife as it is a red light district. Whilst I wouldn't go swearing in Japanese at any of the well-heeled, little-finger-deprived yakuza that hang out there, they will seldom even look at a foreigner, never mind cause them any trouble. Too much hassle.
Roppongi is slightly different in that it's really a foreigner hotspot - plenty of dodgy bars and clubs, particularly of the 'hostess' variety, which will invite you in and charge you hundreds of dollars for a drink and some chat time with a Japanese hostess. Technically you're paying for the drink, but argue at the price and expect a rude exit from the establishment. If there were anywhere in Tokyo you shouldn't go, it'd be here at night - but only because it's not very nice, as opposed to dangerous. During the day Roppongi has some rather spectacular architecture, and is home to a bunch of Western and Japanese companies, as well as a ton of embassies. Not really a hellhole by any standards.
As with any country, common sense rules. Make sure to keep your valuables in a safe place or tucked away on your person. If you're dangling a wallet full of holiday fun time yen as you saunter through a train station it's more likely someone will try to help you and point it out, but still, it's better not to in the first place. Don't leave luggage lying about either - it won't get stolen, but the police don't like that sort of thing in case it's something more sinister.
Long story short - at no point in Tokyo will you ever cross some boundary from 'nice part crime-wise' to 'holy christ it's the Baltimore projects'. Those boundaries don't really exist.