No running. No jumping. No minecarts. Some developers in Uruguay have been making a game called Bullet Boy that seems to be exclusively based on the barrel levels of Donkey Kong Country (perhaps the most fun parts of those games?). Don't settle for a trailer. You can play a short demo in your web browser. Right at this link.
The guys at Pomelo Games tell me that they're aiming for a mobile release—iOS and Android—for July or August. They'd like to do a PC or console version, too. You can find out more at their dev blog or hang out on their Facebook page.
I think their demo "trailer" is fun, has good art and shows a ton of potential. I hope you enjoy it, too. It takes all of three minutes to play through. Have at it!