We are not native speakers of English. So we don't sound like one. And we most likely never will, if we don't sound like one by now. But that's OK. There's no need.
There are more non-native speakers in the world than there are native speakers. So we are more likely to be using English with other non-native speakers than with native speakers (unless you immigrate to the US or get a job at the British Embassy or something). It is certainly the case with me. Few of my friends with whom I use English are native speakers (except my colleagues at school). I attend an international church where English is the practically the only language used, but very few in the congregation are native speakers of English. They seem to understand my Japan English without much difficulty.
It is probably not true that native-speaker varieties of English are the most widely understood. There are quite a few studies which found out that native-speaker varieties were actually harder to be understood in international communication. And I have attended many international conferences and meetings where native speakers were the only ones having trouble understanding and being understood. It's not hard to guess why. Many of them are so used to speaking in English that they speak so fast, and they don't enunciate clearly. And they use many local idioms and slangs that may be common in their own countries, but not outside. Some of them probably put the blame for communication breakdowns on the fact that our English is "not up to the standard". We know from studies that such an attitude interferes negatively with one's ability to understand.
Aside from being understood more easily in international communication, there are other benefits to sounding non-native. I feel more comfortable and psychologically stable when people can tell from my accent my cultural background. This way, I don't have to try to pretend to be someone else. It is tiring to sound like a native speaker. It is even worse sometimes even imitating native speakers' mannerisms, facial expressions, etc. (I think many fellow non-native teachers agree with me.) It is a futile effort. And it makes me feel stupid. Besides making me feel as if I was a copycat monkey (物まね小猿) in the English-speakingdom, it actually does make me sound more stupid, because I can't pay much attention to contents, grammar and lexis when I have to pay so much attention to native-like pronunciation.
Another thing is that it is dangerous nowadays to be mistaken as a native speakers of English in certain countries, especially in the Middle East, etc. But this is a sensitive issue, so I won't talk about it here then.
OK, so if we don't need to try to sound like a native speaker, how can we raise our success rate in international communication with other non-natives? Here are my thoughts.
From my experience, it is important to pronounce the words clearly. You should almost sound like "spelling pronunciation". This will certainly slow down your speech. You will also sound less "native-like" and more "foreign" to native ears. But that doesn't matter. People from different countries will understand you better.
Studies found that "syllable-timed" accents were easier to understand for non-native speakers. Syllable-timing means you give almost equal stress to every syllable in a word. So you sound like "cohn-grah-choo-lay-shee-ons" when you say "congratulations". Many proficient non-native speakers of English have syllable-timing. Yet, syllable-timing is one thing that will make you sound definitely non-native. You may feel you are less "cool" when you sound less like an American. But you have to realize that you will never sound completely like an American no matter how hard you try anyway. (Unless you have some special talent, that is. And if you do, you should perhaps consider becoming an actor.)
I think one of the reasons why "spelling pronunciation" or "syllable-timed" accents are easier to understand is because, especially here in East Asia, we learn how to spell and read English before we learn how to speak it (if we ever do)! When I was learning English as a young student, I always felt frustrated that English words never sounded like how they are spelt when pronounced by native speakers.
Another important thing is more of a psychological factor. It is called accommodation. It means you have to accept that people have different accents, and be willing to try hard to understand people with "unusual" or unfamiliar accents. This can happen only if people are constantly exposed to people with many different accents. The opposite happens when people are exposed only to limited varieties of accents. For example, many Taiwanese students are familiar only with a kind of American accent. So they may mistakenly think that is the only "standard" accent of English. And when they come across someone with a different accent, they naturally find it difficult to understand. This is usually because they subconsciously blame the other person for not having what they think is the "standard" accent. This causes them to give up efforts to understand too easily. The problem can be solved easily by constantly exposing learners to different accents.
It is encouraging to know that in many countries, like the Philippines, people are "functionally native" in English. It means that they use English among their own countrymen in certain domains. Even though the way they use English may be different from that of, say, Americans, and occasionally get laughed at for that, it nevertheless functions perfectly well as a lingua franca in those domains.
I often use English with my fellow East Asians including Koreans and Vietnamese, because it is often the only language we have in common. And it is very helpful that they find my Japanese rhetorical style and cultural references very similar to their own. The communication would be much more problematic if I spoke with an impeccable British/American English. Well, Asian Englishes is another topic so I will write about it some other time, then.
