When I log in I get black boxes notifying me there have been replies to my posts - so many they overflow down he bottom of the screen - but they are all for posts in the previous migrated forum, mostly years old.
Will this continue until all the old posts have alerted me?
Also, these have a tendency to disappear before I can look at them or delete them or whatever.
Is there a way to call them back manually (or see similar data in some other form)?
It'd be nice to know which topics I've posted to have had additional posts, but I don't think I see
any such data for the current (new) forum topics...
On the top right of the screen is a light gray box with the search and your user name. When you click on the link "replies" you get a page for all replies on your topics/topics you have posted in. There should be a button "Mark all as read" or similar. Press this. After you've done that the list will be cleared and the black boxes will disapper.
You can also go over "unread" and then "Mark all as read". No matter that you press, both lists ("unread" and "replies") will be cleared.
Jan
P.S. It's quiet funny to press mark all as read under unread, and come back after half an hour our so to see how many new post there are 
EDIT: Under "replies" the same info is shown as in the black boxes if you don't have the time to see what is in the black boxes.
After I press "Mark all as read" I still get those stupid notifications pop up! Even telling me that I replied to posts from the old forum. Really annoying. They also popup and random times when simply browsing through the new forum.
Must say I'm NOT liking this new board... 
The site is still a bit buggy. They've said they are working on it. I'm just ignoring them for now.
Mark as read only marks what is visible listed on a pages worth. Keep doing it and you’ll get through more pages until they run out.
For now, I'll accept the bugs
If they're still there in, say, 2 months? I'll be a little more upset
you sure 'bout that?
I thought it marked an entire (sub) forum's worth as read
If I chose "unread" and then "Mark ALL as read"(or similar) ALL post in every sub forum and sub sub forum are marked as read for me. If that's not true for you, I can't help.
JanD