My understanding is that Fr.Ephraim is on the bad side of a lot of the greeks (or at least the modernizing/secularized parts of the GOA) because he has brought Athonite monasticism to North America. In a healthy situation, this would be no crime.
It goes without saying that there is a lot in traditional Orthodox monasticism which goes against the grain of worldly expectations on what the "good life" is. Obedience, poverty, placing one's hope totally in the "world to come", etc. Such things are particularly unsettling to modern western/American civilization, which is quite materialistic and secular.
Sadly, a lot of this seculirizing tendency is present in the GOA, and this is where a lot of the complaining is coming from. In part, I think these people are even worse than the non-Orthodox mainstream American, in their unreceptivity to authentic, traditional Orthodox Christianity. It's no surprise that these are also the people most itchy for reunion with Rome - it seems to me that many of the Greeks in the USA have been trying to be looked at as being "respectable" and "noteworthy" in American society, no matter what the cost. In short, it smacks of an inferiority complex. It's like they'd be embarassed to have their priests walk around in traditional priestly garb, with long hair and scragly beards... let alone have to say that their son decided to run off not to become a doctor or a lawyer, but a monk.
While I think it's fair to say that there have been growing pains in Fr.Ephraim's monastic establishments, the treatment he and the monasteries themselves are receiving are signs not of problems in the monasteries so much as an unhealthy American Greek Orthodoxy.