Networking problems can be a little bit of hit and miss when troubleshooting. My first guess is that this is not related to your wireless router, but to the way the Apple devices are renewing their IP address from the router (I presume that you are using standard settings and are handing dynamic IP addresses via DHCP). I say this because you mention that your Ubuntu laptop has no issues and because all devices continue to show an active Wi-Fi indicator.
My recommendation would be to first make sure that there are no IP conflicts between your devices. First, turn off the Apple TV, iPad and all Apple devices except for the MacBook Air. Then use the MacBook Air as you normally would to find out if you still loose your connection to the Internet after an hour or so. If you don't loose your connection to the Internet, then turn on the next Apple device and pay close attention to who is the first one to loose connectivity and when.
I have seen this problem happen to me whenever I use the my Clear Wi-Fi hotspot when I am on-the-go. I will get messages from the MacBook (not the iPhone, although I know they both are having the same problem) telling me that the IP was lost because another device is using it. This problem does not occur when I am at home using the Wi-Fi network served by an Apple AirPort Express rotuer.
I highly recommend the AirPort. However, if you want to keep using your existing router and you confirm the IP conflicts I suspect of on the paragraphs above, then I suggest configuring your router to not hand out IP's with DHCP, but rather to have your devices use static IP's and see if that helps. It is a little bit of a pain to do so, but it may be the only way your devices can co-exist.
Hope these thoughts help. Good luck.