Oh, the US has a huge variety of accents.
My own accent is pretty much RP with some Westcountry colour and has never been an issue with students: it's pretty much indistinguishable from the English they hear on the CDs that come with their textbooks.
Generally, non-native speakers don't have an "ear" for different accents as long as they are not too far removed from the standard they've learned. I doubt if you could tell the difference between a Bavarian and a Berliner speaking German unless it was pointed out to you.
However, if you were learning German and heard a broad Swiss German dialect, you wouldn't have a hope.
The issue is, however, less one of your students understanding you, and more of what they learn from you. If your students end up speaking nothing but Geordie, they would flummox any American; and also be flummoxed by American speech.
Be reasonable. The difference between, say, the RP pronunciation of "cake", and its pronunciation by an educated speaker of the Yorkshire dialect, will pass unnoticed by foreigners (Germans, for example, have trouble with the PR diphthong and hear it as the pure "Yorkshire" vowel sound anyhow) and be of no consequence to any native speaker (Americans had no trouble following the speech of Daphne in Frasier, although British viewers couldn't help noticing that it sort of drifted back and forth between Manchester and Scarborough). However, I have used old episodes of Last of the Summer Wine in lessons occasionally (as a special treat), and my students have always been grateful for the (English) subtitles.
Speak clearly, avoid actual dialect, and try to sound like an educated person from wherever you come from. But don't try to sound like Harry Enfield doing one of his spoof black-and-white documentaries.