I speak fairly fluent Hebrew as a second language, and I have a UK English accent. I want to know what pronunciation issues generally would be an issue for English speakers and would sound strange/be obvious to a native speaker aside from the guttural r and kh sounds which I have a fairly good grip on.
I'm no expert in Hebrew, but I did find this extremely helpful YouTube playlist of 6 videos that directly addresses difficult pronunciation sounds in Hebrew for native English speakers. The videos are also very recent, as the last one was posted in September 2015.
I have exactly the same problem, except that I've got an American accent.
Besides for the r and kh (which I still can't nail the difference between Israelis' pronunciation and us American Jews' pronunciation), there's the obvious difference of vowels - if you start pronouncing the patach (short a) sound like a Brit, the taxi drivers are going to rip you off by at least 50 shekel. Beyond that, just remember that most places in Israel read the Taf and Saf the same way, and you should be fine. :)
There are two big issues outside of 'r' and 'kh' sounds:
- Vowel length. English speakers tend to have much longer vowel sounds than Hebrew speakers. Shorten your vowels.
- Another aspect of pronunciation that many don't think about is the speed at which you speak. You could have perfect ר and ח, but if you speak at a snail's space, then you will still stick out as a foreigner.