I really appreciate those foreigners who do their best to make their accent in Russian sound as native as possible. As I noticed from my personal experience, it can even sound cute - English speakers often can't pronounce И (i) and Ы (y) differently, so they usually pronounce И instead of Ы, and it sounds softer and cuter. Some of them (whom I met at least) also tended to speak Russian slower and quiter than they do in their language (they probably needed time to think a bit). At least I heard some learners of Russian who talked like that and it sounded cute and a bit like baby-talk. But of course, they don't sound softer in anything, because there are a lot of other sounds that can be only hard in English (and in many other languages as well), but they have a soft pair in Russian, so there's an opposite situation, when they can (or do) pronounce these sounds harder.

BUT I spoke about the sounding of those guys who do their best and try to imitate our sounds and speech.

As the heavy Russian accent sounds terrible for English speakers, the same goes with the heavy English accent in Russian, and we also sometimes make jokes of it. Especially American English, it seems like they open their mouth while speaking even more than British do, or just in another way, and in Russian you don't need to open your mouth almost at all...

I'm sorry, but I do get irritated by English speaking foreigners who try to speak my native language. We have a Spanish sounding "R" and it is very rare for an English speaker to get it right. Some do, but it is rare.

And since my name starts with "R", it does irritate me when they get it wrong :P

So yeah, "R" is my pet peeve.

But that is the only thing. I do appreciate anyone who tries to speak, don't care about the accent, don't care about (small) mistakes.