I have saved up the dividends which my investment in an Islamic bank yielded.

Answer

What is the basis for your doubt? Zakaat is payable on cash, gold, silver and stock-in-trade. So what has prompted you to ask this question? Perhaps you do not know the answer. The answer is simply the following Hadith of Rasulullah (sallallahu alayhi wasallam): "Sin is that which agitates your conscience." Be grateful to Allah Ta'ala for keeping your Imaan alive. The dividends acquired from these haraam capitalist banks called 'Islamic' banks' are haraam. It is Waajib to get rid of such haraam money. Give it to the poor or pay your income tax, rates and similar other zulm taxes and fines with it. Zakaat is not payable on any haraam money.