The National Art Center
This center is unique among Tokyo art museums in that, instead of maintaining a permanent collection, it is a revolving door venue for art exhibitions from around the world. It has the largest exhibition space of any museum in Japan and can hold up to ten exhibitions at a time although it's usually not completely full.
While some of the shows require admission there are usually a few free exhibitions at any given time. The building itself is worth exploring for its sleek architecture, public spaces and restaurants perched high on wooden pedestals. Check their website for a rundown of what's currently showing.