Yeah, the "blindness" to sounds that don't exist in your native language is funny, and usually the best way to figure out where a non-native speaker is from. If you can't even hear the difference between "sheep" and "cheap", or "joke" and "yoke", you have no way of pronouncing it right in the first place!

Likewise, I can identify English with a Danish accent pretty quickly, but I can't for the love of me speak it, because I'm not good enough at Danish. But a dead giveaway is that Danish realizes 'R' as [ʁ], and many Danes pronounce it the same way when speaking English. :-P And there's something with the vowels that's just weird that Danish people do... I can't put my finger on it.

(I have the opposite problem, I can't make that 'R' sound at all, which is why my Danish sounds pretty funny to native Danes.)