Can I print my own Lego bricks? Assuming I've 3D design (or I've created one) which looks very similar to Lego bricks, I am allowed to 3D print them for my personal use?

Do I need to obtain some permission to do so, because of some patents? Or how does it work?
The patents that cover Lego bricks have expired, so you are free to print bricks using the same interlocking system. You are even allowed to offer such prints commercially.

What is not allowed (and a violation of trademark law) is to call them "Lego bricks" or use Lego's logo. "Compatible with Lego" on the other hand, would be fine.