As a 7-year DC resident, I can say confidently that your friends are wrong. The Alliance for Biking and Walking ranked DC as #2 best city and DC also ranks among the top for bike commuters and in general. It is also pretty safe at all hours, but like any city, keeping your wits about you will help you avoid trouble. In addition to what others have said regarding cycling, I'll just add this: if you ever are going to take eyes off your bike, even for a minute, make sure it's locked with a U-lock. DC has a pretty severe bike theft problem, but thieves tend to go after the low-hanging fruit, by which I mean bikes with cable locks. Bike parking is abundant in the city, so that's not an issue.

When possible, I'd suggest leaving your own bike in the hotel or wherever you're staying and checking out Capital Bikeshare. It's a cheap and convenient way to get around the city (but it's not meant for long rides, just short trips) and you never have to worry about locking up your bike, you just dock at the nearest station and you're all set. There are something like 165 stations in the DC area (DC, Arlington, and now Old Town Alexandria) and 1600+ bikes, so you have to go out of your way to not be close to a station.