I like the Aldi I used to go to in Kansas while working there. Cheap, clean and no one bothers you. It actually was a better experience than Walmart, even for this American. They had cool, better carts too!
Trader Joe's rocks. Evidently things have turned around for Aldi since I remember hearing about them having problems with stores in the southern part of the US. Honestly, that's been 8-9 years ago now, so it would appear I was wrong about their current successes!
AFAIK Aldi is expanding in the US. Also, Trader Joe's which is hugely popular with the younger crowd (at least here in Chicago) is owned by the same company.