Yes, I use a money belt when travelling. I've only had trouble in Beijing and Italy, and both times it was grabbing visible cash rather than more sophisticated pickpocketing or outright mugging. Even so, I think it is good advice as losing your passport is a major PITA, and losing all your cards is almost as bad.
For the record, I use the money belt for my passport, a spare set of cards (credit and debit/ATM), my emergency cash (dollars) and whatever local cash I don't intend to spend that day. My wallet has one days worth of local cash, plus one or two credit cards. It only has a debit card if I am on my way to an ATM or I am returning from one.
How does a passport fit in a money belt? The ones I remember seeing were only big enough for bills and cards.