NCRanger
Bettman's Enemy
There is a free shuttle bus available from a restaurant called Nemo's to the new arena in Detroit. Another option would be to park cheaply at Wayne State University and ride the new M-1 rail system to the arena -- not free, but cheaper than the $40 - $45 parking near the arena (it's $20 if you park on the other side of the freeway and walk several blocks).
The first two options above add quite a bit of time to whatever your normal commute would be to the arena. When the Wings played at Joe Louis Arena, there was cheap (under $10) parking available a few blocks away, and the Nemo's shuttle bus was also an option.
I've been to downtown Raleigh quite a few times, and there is quite a bit of free parking available (we've never paid to park). Never been to a game there, though, so I'm not sure how far you would have to walk to the arena or how early you would need to get there.
The problem in Carolina is that unless you know there may be free parking across the street at the Fairgrounds, you are forced into the arena lots.
I haven't been there in years, and I do know there is a newly developed area. I don't know if parking over there is free.