I'm an Indian born American resident.
My favorite accent is by far General American (the dialect of most American cities outside of the northeast). Even in Texas, GA is what you mostly hear in the cities. Because I hear it so often, I fail to appreciate it as much as I should, but I still do love it. I'm also quite fond of many Boston/New York/New Jersey type accents, but I don't know the differences between them well enough to specify.
My least favorite dialect is by far Cockney. I don't consider it inferior or anything like that, it just doesn't please my ears. Received Pronunciation is okay, but still less sonorous than most American dialects.
I don't dislike Australian English, but I also barely understand the colloquial speech. I'm a cricket fan, and while flipping through the channels one day, I found some Australian domestic cricket. I swear, that for five minutes straight, I had no idea what the commentators were talking about. Even the words didn't sound English.
I've found that virtually all of the Americans I know, of all races, find "British" (I'm assuming they're referring to RP) accents attractive and my indifference to said accents weird.
Whenever a non-native speaker of English is speaking, I take great delight in analyzing their speech for traces of phonetic influence from their mother tongue :]
As a result, I always have great fun, but often forget to pay attention to the meaning of their words :[