The major sites are in bike friendly areas. Around the Mall and along the river is easy riding. Bike racks at museums and most of the monuments.

If you have a pack or panniers on your bike, be aware that there are a few areas that have higher security, such as around the Capitol and White House, where you may be subject to search. And i fyou are in one of the high security areas, don't leave packs on your bike and walk away. Unattended packs may be mistaken for a bomb attempt and you may lose your bike to a bomb disposal unit (ie. *boom*) Suggest a backpack. It will still be searched but you can take with you and leave it at the coat check while you are in the museums. What I do is unhook my TransIt pack and take it with me.

You have the right idea. Bikes are a good way to see the city.