I have a mighty mouse on my iMac. I have no trouble right clicking with it in Dominions. I tend to play in windowed mode, but I did not have any problems with full screen either.
Sometimes I find that right-clicking with the mighty mouse can be picky if you are not holding the mouse just right, which is a bit annoying.
My trouble when I first got a Mac last year was patching Dominions. I needed to change the permissions. Post if you're still having trouble with that.
I agree with PashaDog. The Mighty Mouse is not Apples best product. Office Depot had a basic $7 Microsoft mouse that worked as well as the MM (didn't look as neat).
Also check out the trick of holding down the scroll ball to drag the map. Does that work? It does with my mouse.
I started using the windows mode all of the time. I have to trick the window to reproportion by clicking the green (minimize/maximize) button. This allows me to see button over messages at the bottom of the screen.
I personally found that the way I hold and move the mighty mouse was just wrong enough that I was triggering the wrong clicks a lot of the time. I ended up turning off the right-click (and the side buttons) and going back to a one-button configuration (using key combos to simulate a right-click).