Just how "cheap" do you want? Since you're an American, you should be familiar with Kentucky Fried Chicken, McDonalds and Denny's. They're all in Japan. As mentioned before, Yoshinoya is also in Japan. Their menu is a little different than those in the U.S., but generally they're the same and cheap. I ate at Yoshinoya a lot during my last trip as I had to watch the cash flow towards the end of the trip.
In Hibiya Park in Tokyo, I treated a friend to lunch at a revolving sushi restaurant and the total tab for the both of us was about 2,900 yen. I thought that was reasonable.
On another trip, in Ueno in Tokyo, I ate at a sushi restaurant and they had a special for around 900 yen (roughly $8.00 and change).
Eating in Japan is not expensive, you just have to scout around.