A chain catcher is designed to stop your chain falling off when you shift down from the big to the small ring. They also help keep your chain in place when you ride over rough terrain and when dismounting and remounting. Chain catchers are an inexpensive upgrade (depending on which one you buy) that offer insurance that is hard to beat.
One of the cheapest on the market is the Deda Dog Fang. I run a very similar chain catcher on my cross bike - just a knock off I found on eBay for $5.
If you want to go more upmarket (and I suspect in weight) go for a K-Edge model. They come in about 8 different configurations including clamp or braze-on and one that sits between the bottom bracket and frame, like the one below.
This one, is a cyclocross specific model that can be mounted higher on the seat tube away from the a flared bottom bracket.
Check out the K-Edge website for all their products.
There are also examples of home made chain catchers using cable ties (is there anything they can't do?)
Cheap and awesome, a chain catcher will save you time and you won't even know it.