I recently moved and I can't connect to peers with Bitcoin or bittorrent with my new internet. I am not using a router, I am plugging a cable directly in. I will be getting a router tomorrow hopefully, but I don't think that's the issue anyway. I haven't tried much of anything, all suggestions welcome.
It is more probable that it would work without the router than with it anyway as you could have had some firewall configured on it. I am assuming it was working with your previous provider and you didn't commit any changes to your OS/PC?
Do you configure your public IP yourself or do you get it from a DHCP server? Do you have a public IP at all? If so, try using some on-line port scanner on your address and see whether your port 8333 is listening for connections or if it is not closed at least. We can go further from there.