I am a native speaker of English (I grew up in Alberta, to locate the accent more exactly) and there are a few issues that I try to pay attention to when I pronounce Greek. For instance, I try to avoid lengthening short vowels in open syllables, and to not voice sigmas between vowels (but I do voice them before voiced consonants). Getting the aspiration right in stops is hard (tau and theta, phi and pi). I'm sure I'm missing some.
What other mistakes should an English speaker be sure to avoid?