My cable company says my router is searching for an internet connection 1000's of times a day and that I need to update the firmware for it to fix the problem or they'll have to shut me off... I've looked and haven't found any update for my particular router, a Zonet ZWA3006... Any ideas?

I would start by doing a reset of the router to factory using its procedure. Most routers have a reset button that if you hold it down for 10 seconds will reset back to when it was new and erases your settings but might be the trick to get it to work again. After you input your settings again, I'd check the router maker's support site for a firmware update and update the router. These updates can have fixes for just such issues.
I have had older routers that failed and will give weird behavior like you described. I just replaced it with an Apple Airport router and have been delighted with it.

update the router firmware
You can do it via Linksys or as many of us have done, use this 3rd party firmware
http://www.dd-wrt.com/site/support/router-database

1. Firmware update may be required for the Router.
2. check your firewall settings and the Wireless configuration in the router.
3. You may have to allow the MAC address for the device that you are trying to connect with to have permission to connect to the network.
4. Verify IP for the the PC ( this is done in the DOS window) at the DOS prompt type "ipconfig /all" this will bring up the network report as to what is and what is not connected.
5. Unplug router and modem from the AC.
6. Reboot the PC.
7. Window is totally different from the router firmware updates.
Windows is downloading it security and web update when you see the prompt. This has nothing to do with the router.

1. check the current firmware updates
2. verify no other devices next the the router interfering with the signal i.e. cell phones, cordless phone, microwaves, TV sets.
3. check your router or cable modem for connection during time the wireless is dropping connectivity.

Cox people are really correct that everything is working from them and from there modem and doing a firmware update is really a good thing to do to correct your problem. For that firmware update you just need to download it from your routers manufacturer like Linksys , Belkin , Dlink or Netgear just go to there web site and search the model of your router so you can download the latest firmware for it.

have you restarted the router? if so, then see if Belkin offers a firmware update. You will need to connect with a cable to do the update.

Have you tried updating the firmware of your router? Updating the firmware of the router will fix that issue. Make sure after updating the router's firmware, hard reset the router. To hard reset it, Press the reset button on the back of the router for about 20 seconds, then unplug the power for 30 seconds then plug it back. Reconfigure the wireless settings of the router. Make sure to change the wireless channel of the router to channel 11 or 1 and not 6 to avoid interference.

Yes, you need to swap the router out.

This is assuming you dont have the netopia in bridge mode with another device behind it.

spin

"Update your router software(firmware)" is a common solution that is suggested for fixing router problems...whether there is a good reason to do it or not. It is possible that your router has a software problem...so it won't hurt anything to upgrade it...and it might help! This Belkin web site has instructions and downloadable firmware files.

Well before throwing it out try updating the firmware. That might help a little. http://search.dell.com/results.aspx?c=us&l=en&s=bsd&cat=sup&cs=04&k=2350&img=True&sum=True&ssum=False&qmp=25