It was a point-and-click adventure from somewhere in the early 90s. The story had kind of an Arabian Nights type of theme, with djinnis and magic. No affiliation with Disney's Aladdin.

Simon the Sorcerer?

That's what first came to mind, though I was thinking of Sierra's original, not AGDi's remake.

Quest for Glory II is the first to spring to my mind too, but it was a text parser adventure, not point-and-click

And despite being the only game in the series never to see the light of VGA it was the one I enjoyed most. I should fire up the AGDI remake sometime...

Wow, quick replies. :)

It's definitely not Simon the Sorceror. Al-Qadim is certainly along the right lines, but I'm sure it was more collecting items and puzzle solving and not running around dungeons avoiding traps. It was also more colorful than Quest for Glory 2 and again, not so much rpg.

Edit:
It wasn't a platformer like Prince of Persia, nor a mystery/horror (DoS). Thanks for the guesses, though.

Prince of Persia 2 perhaps?

After doing some searching on youtube, it looks like it was Al-Qadim after all. The bit with the checkerboard puzzle in the desert and the two djinn on the mountaintop looks very familiar, even if the rest of it doesn't. Thanks a lot everyone!