I was going to start to maybe try to answer your query earlier as I also own a DNS-323 NAS and it was connecting fine when I first set it up. I don't leave it on all the time though and I hadn't tried to use it since I finally got my DIR-825 Router upgraded to firmware 2.05NA. I had tried 2.03 and 2.05 when they came out but found they limited my upload speed by default to 1MB for some reason. I didn't know about the setting until recently that would fix it because D-Link support never knew what to tell me what to check. They had me jumping thru hoops and not pointing me in the right direction. I have a thread started about their lack of knowledge of their own hardware and software as it pertains to the upload speed issue.
After reading your post, I started to offer you things to check but then thought about the fact that I hadn't turned on my DNS-323 NAS since getting the 2.05NA firmware for the DIR-825 Router to work for me as far as my upload speed issue. I turned on the NAS. The Easy Search Utility wouldn't find the NAS. I reset the router to factory defaults and it would see the NAS but it wouldn't connect to it and map the drives. Eventually it wouldn't see it at all without rebooting the NAS. It never would connect. I flashed the DIR-825 back to 2.02NA and guess what, no problem connecting to the NAS and mapping the drives.
D-Link is a waste of time at this point. Unless you FULLY understand what every last setting should be set at to get something to work that D-Link broke with their last 2 firmware updates then you are stuck Googling your butt off looking somewhere else for an answer. Their support in this forum is NON-EXISTANT. Their support ticket is a waste of time because the people answering you have no clue as to what to do. Like I said, you have to FULLY know and understand the effects of EVERY setting for the DIR-825 in order to have your network function because D-Link has screwed up things that use to work as it was by default when using 2.02NA.
If you are using 2.03NA or 2.05NA, I would suggest downloading 2.02NA and flashing that on your router. I bet it fixes your issue with connecting to your NAS. Do me and everyone else a favor. If you do flash back to 2.02NA and it fixes the problem, PLEASE post back here in this thread so D-Link maybe will see it and quit making excuses that their firmware isn't the problem.
All the above suggestions is if you are using the North American version of the DIR-825. That is what the "NA" stands for. It also must be hardware version B1. The hardware version will be shown on the bottom of the router in the lower left corner of the info tag. Again, PLEASE post back if you do this and it fixes the problem.