Claes,
I've been poking around your site a bit today, and really am not sure what's going on there.
First, when I went there, the home page's first article had 1 Like (I think), and the rest had 6. I'm assuming these last articles were all 'linked' like you said. I was not logged into Facebook at this time.
Then, about 30 minutes later I came back to the site, logged into Facebook, and all the Like buttons looked like they had 0 Like's (no number next to them). On clicking on them, the counter would increment to 1, but it was not posting to my wall and if I refreshed, the button had reset itself.
I'm not sure why your site isn't liking Like button clicks. We'll continue to investigate, but do can you let us know:
1) Was it ever working on your page, and just broke?
2) Has anything changed recently (if it was working before)?
3) Can you try with a different template to see if you have the same issue? Not sure why it would be template related, but that's the best guess I have for right now.
Just so you know, when we've seen "linked" like's, it's generally because the final article page that's being liked usually has an Open Graph URL tag on it that's pointing to another page. Therefore, if on your blog, you have 6 articles.. then, each of those articles each has an Open Graph URL tag that's pointing back to the home page, all likes would actually count for the Home page, not the individual articles. This doesn't look to be the case on your site, but if you had a default Open Graph URL tag configured in JFBConnect, that could have been the cause of the problem. As Facebook re-scans your site, it should clear itself up.
Hope this helps, and please let us know if you find anything.
Alex