Less than perfect English an obstacle to getting teaching job?

[theofan]

I have a couple friends with theology PhDs who are non-American and speak English as a second language.  Their English is not perfect: they make grammar and pronunciation mistakes here and there, their vocabulary isn't too sophisticated, and they definitely have an accent from their native language. They haven't had much luck getting hired as teachers here in the US. Do you think their English is an obstacle? Would an employer be turned off by someone they can't understand perfectly? Can publications make up for imperfect English? Another friend of mine thinks it's a big factor, so asking your opinions.