I entered Goma/DRC from Gisenyi/Rwanda about 2 months ago, but didn't venture further inland. I paid some "extra money" for the visa on arrival (not sure if this would be valid in the whole country). No serious problems around Goma these days.
Recent rebel activity has been reported up north at the Sudanese border (LRA) and near Bukavu (FLN, and another group I don't remember)
As far as I understand (and as written in the previous post), you could head south via Mbuji-Maji. The roads are probably ok with a good 4x4, patience and experience. I wouldn't worry about the permits, as everything is just a question of money in that country.
I assume that the northern route via Kisangani has not been travelled lately, but I am sure it could be done, if you are prepared to cut trees, build bridges, etc.
How about crossing to the west coast on a southerly route through Zambia and Angola, via Saurimo? The only problem is getting the Angolan visa.