The discussion board is not very advanced in 2010 version, they have improved it in 2013. The discussion board does not permit out of the box anonymity.
The only feature that allows to be anonymous is the survey (although the site owner can switch it off anytime and see the names of users), which is not made to discuss other people's replies. They could, but it is not easy to navigate among replies of other people and one would not have a place to comment.
Per experience (and I am not talking about technical things but basic human nature) one thing is to request feedback on more important policies of the company, which for many employees is considered risky (we all know how the employer might treat a personal opinion not in line with his own), another thing is to receive feedback on the use and ways of sharepoint which does not necessarily mean critique towards the employer and should be considered mere improvement to current hardware/software/policies.
I would not architect something more complicated since the signal you are sending out is that merely talking about sharepoint and its use inside the company is something dirty and something to be careful with. This sends out a negative message that might get in the way of your user adoption strategies.
If the business culture is so bad that anyone would feel ashamed to post an idea for fearing to be judged as silly or ignorant, use anonymous surveys. If the culture is more relaxed and allows participation, free of prejudice, then use the message board, which has the advantage that keeps everything transparent.