I personally like the following, some of which may have been mentioned above:
The Netherlands Carillon/Iwo Jima memorial - you get the classic Iwo Jima statue/flag memorial, plus as an added bonus you get the classic postcard shot of the Lincoln Memorial, Washington Monument and Capitol in one shot. Located in Arlington a small walk from the metro station. Easy to find.
Korean Memorial at night. The statues are all eerily lit and they look like they've come alive when all but the statues is fading into the darkness of night. Downtown, just south and east of the Lincoln Memorial.
The WWII memorial, at the east end of the Reflecting Pool. Also, you can get good shots of the Washington Monument around this area coming from the Korean Memorial.
Sunset over the Capitol from the steps of the Supreme Court. Great colors.
I don't know if it's picturesque or not, but the Old Post Office Tower is the second highest point in the city and you don't need tickets to get up it. It's a very good view of everything, with only the Washington Monument blocking a very small portion of your view. Located only a few blocks from the Whitehouse or Washington Monument.
Good luck with your shots. I lived there for just over four years and miss it a lot. It's a very photogenic city, and I'm lucky to be headed there during July 4th weekend to visit friends and hopefully get some of my own shots.