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I'm an international, teaching at an American University, and I have an accent, which can be strong at times. I'm working hard at it, have always been, but still, the accent is there.

This has never been a problem in my career as a researcher because I was most of the time working with senior researchers and grad students. And even if they were native, they've never expressed any concern about my accent.

I'm sure that several times they didn't understand a specific word I was saying but they catch the meaning from the context. Sometimes, people asked me to repeat in an informal way by simply saying "what's that?" or "say it again".

Last year, when I started to teach undergrads, which are unlikely to have experienced different accents, I had some issues.
Some students asked me to repeat and I gladly did. My concern is related to those students that didn't ask, either because they are shy or because they thought that by asking I would be offended.

This next semester (yes, I'm going to teach in the summer) I'm thinking to tell my students in the very first minutes of the first class about my accent and tell them that I'm totally fine if they need me to repeat some words.

I'm not sure about it, because I've read that the first 6 minutes with a new class are the most important ones, and by showing this "weakness" they will conclude that I'm not capable, knowledge-wise, to teach them.

A mentor of mine told me I should talk openly about my accent, and add that in addition to English, I fluently speak two other languages and can understand two others. Trying then to make a balance between my "weakness" (my accent when speaking English) and my knowledge with languages.

Accents are tricky, especially in large lecture halls where students hear you over a mic, and can't see your lips. It would go a long way for your undergraduates to say something like:

As you've noticed, I have a strong (native language) accent in English. I know sometimes that makes it hard to understand certain words. Please don't hesitate to ask me to repeat something.

This would help with students who are shy or don't want to offend you. If you like, you could mention the other languages you're familiar with to encourage speakers of those languages to connect with you. This might impress mostly monolingual Americans into being more understanding.

A difference isn't a weakness. Don't think of it that way. You may find difficulty in communicating because of that difference, but it is just that, a difference. People in the US from Alabama and those from Boston speak English with different accent and different idioms. Eventually we get used to hearing a different accent so the effect lessens.

In fact, you and I would have a lot of trouble communicating since I am quite deaf and technological solutions are only partial. Even it the best of situations a speaker sometimes needs to repeat or - better - say an equivalent thing with different words.

But the first few minutes is, IMO, a good time to introduce yourself and how you speak. In fact, it can be fun if you "put on" an extreme version of your accent, just to show the range. Cockney slang, for example.

I agree with others that it would be good for you to address your accent. In fact, I would suggest that you add a statement to your course syllabus offering help and encouraging students to speak up if they can't understand you.

As an academic advisor at a university that has a high first-generation population, I often talk with students who are struggling to succeed in a class where the professor has a strong accent. When I ask whether they've spoken with the faculty member about the issue, they often reply that they have been too shy or thought it would be disrespectful to do so. Having a written statement in your syllabus offering asking if they can't understand you (as well as speaking about it verbally) may then make them feel empowered to help themselves by asking for help! :)

You certainly can, as other answers have suggested. But if you get the feeling - over one or two semesters of teaching - that this is impeding your work in class, don't consider it beneath yourself to try to get some professional assistance in practicing your diction. Very often, people are taught foreign languages with almost no training in pronunciation and diction - which can be quite challenging depending on your native language. (As a personal example - I find the tonal pronunciation requirements in Chinese horribly difficult!)

Also note that even if you tell your students it's ok to ask you to repeat yourself or speak more clearly - some students would still feel embarrassed or that it's out-of-place for them to do so; plus, after asking that two or three or four times, more students will begin feeling embarrassment of making such repeated requests.

If it's not that bad then forget everything I said :-)