This is a rough problem that I've had myself, so here's some info to get you started.
Actiontec MI424-WR User Guide
Here's a URL to someone's blog who claimed to get it working, but the screenshots are missing so it's a little difficult to follow.
I would post that as an actual link, but ServerFault doesn't like me as a new user so will only let me post a single link right now.
Basically, his claim is that the Firewall Filtering rules are blocking some of that return traffic and what you describe is exactly what I saw. Traffic would come out of a private network, hit the desired server(s) and they would respond in kind by replying through the gateway which was the Actiontec router. Then nothing came back through the gateway.
I played with the Firewall Filtering rules quite a bit but got frustrated because the documentation in the User Guide isn't entirely clear on what is what, nor does it explain what the Filtering Rules contain by default. Even at the most open setting, they seemed to be in place and preventing some traffic. If you make a mistake, it's possible to completely lock yourself out of the router (because it's filtering your traffic) and then you have to do a factory reset to get it working again.
In the end, the solution that I found to work was to set the Actiontec router into Bridge Mode. To do this, log into the Actiontec router and go to "My Network" and select "Network Connections" on the left. Then click on the "Advanced" button at the bottom of the screen to see all of the network interfaces on your Actiontec.
You will need to release the IP address from your Actiontec router so that it can be picked up by your own router. Otherwise it won't work. To do this, click on the appropriate "Broadband Connection" link, depending on whether you're set up using Coax or Ethernet. (On my router, both are connected, but only Ethernet is in use. I understand that for the most part, FIOS uses Coax instead).
Click on Settings at the bottom of the screen, and then you should find an option to Release the IP Address. Do this, and then change the Internet Protocol to "No IP Address". Click Apply and accept the changes, as needed.
Now that the IP address has been released, you need to change the router into a Bridge. Under the connections, select "Network (Home/Office)" and then click on "Settings". You'll see the word "Bridge" near the top of the screen and several network interfaces under it. To the left of the network interfaces there are some checkboxes. Check the one next to the appropriate Broadband Connection you're using (Coax or ethernet) and also check the box in the STP column. Click Apply and accept as needed to get back to the main section.
If you haven't done so already, disable the wireless access point. Also, go into the Firewall settings and disable it as much as possible with minimal security. The reason is that the real router you're going to use should be the one taking care of everything, not the Actiontec.
That's about it. It really sucks to have to use this as a bridge, but it did work. You might find another way to bypass it entirely, but I've been having this problem for quite some time now and this was the first way that I found to get the traffic to route properly. The firewall filtering rules would probably work, but they're a real pain and the documentation isn't much help.