Should I be honest and tell my current employers that the reason I am
leaving is because of there nonchalant attitude towards my role? Or do
I avoid burning bridges by keeping it all genial and making up some
other, unrelated reason?
I usually suggest taking the high road, and giving only generalized reasons for leaving like "I really loved working here and learned a lot, but this is an opportunity I couldn't turn down".
Weigh the options, and the likely outcomes. You have to decide what good can come of being brutally honest, versus what bad can come of it.
The best that you could hope for as a result of being very honest is perhaps sympathy, or perhaps a better investment in your replacement. None of these are very likely, and none benefit you much (if at all) in the long run.
On the other hand, by being very honest, you could burn bridges with your manager, your co-workers, HR, and others - any of whom could be in a position to aid in hiring you into some other company in the future, or providing a good reference.
And trashing the company you are leaving is never a good way to exit. The folks you are leaving behind still have to continue to put up with whatever it is that drove you out - at least for a while. Don't try to make them feel foolish for staying. It's a small world - stay professional to the end.
Most exit interviews are just formalities anyway. When you are asked "why are you leaving?" HR reps seldom actually care if you are giving a really honest answer - it's just something they are required to do.
An exception to this would be if you are working for someone you trust completely, perhaps a manager who is also a long-time friend. She/he may actually be interested in your insight. These situations seldom happen, but it's possible.