Irish people, especially in the Republic, would be very upset to hear their accent described as British.As others have said, there is no neutral accent. There are basically two major categories of accents, broken down into many different varieties. British English is one, and American English is the other. In British English, the largest variations are Scottish, Irish, England proper and Australian. They all share more of a soft, flowing pronunciation. There are many different regional dialects and accents through each of those. American English does not really have any related country accents, outside of Canada and countries that speak American English but have accents from their other native languages. American English can sound harder, with more defined "cuts" between letters.
Typically, with American English, the West Coast (California, Oregon, Washington, etc.) is seen as having a "neutral accent". This is fortunate for people learning English, because almost all actors in Hollywood movies have a neutral American English accent, unless the movie calls for a different accent (such as Southern, Northeast or Mid-west). You could try watching some movies and working on imitating how they speak. Just be sure that they are speaking how you are accustomed (from language learning CDs, class, etc.), or they could be speaking with a regional accent. But most will be neutral. That is, of course, if you want to try to cultivate an American accent.
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