I'm currently developing an idea for a conquest-style board game similar to Risk. I was wondering if I would be legally entitles to use some game mechanics that appear in that game so long as I don't use Risk's title or logo or anything.
Game mechanics cannot be copyrighted, so using the Risk gameplay ideas should not be a problem. Obviously, you can't use the name Risk because that is trademarked.
That being said, I am not a lawyer. You should avoid copying exact rules or procedures from Risk as you risk (pun intended) running afoul of copyright issues.
Yes. There is no harm to use game-mechanics of any one. Even a good mechanics may become a complete genre on which world can run after for at least 6 month :)
So in short, YES you can.