How do I get my iPhone to recognize my new Linksys router.

How many wired and wireless computers or devices do you have?
How many are conencted to the wireless network?
How many are actually able to access Internet through wireless connection?
If your wireelss network is secured then make sure that you have the correct wireless network key.
If nothing is working then I think you should try to reset and reconfigure Linksys router.
Create a new wireless network and then try to connect to the new wireless network
See Linkssy router setup

just confirm if the password you have for internet is correct or not.
Now do the manual security settings in your phone same as of router. Like security type & password both.
check for the phone wi-fi, and use it.

Check the wireless settings of your router. Make sure that Wireless SSID broadcast is enabled on your router.
If other wireless devices are able to see the network and only Iphone is not able to see the network then it might be the issue with the Iphone.
You can try changing the wireless channel on your router. You need to open the setup page of the router using 192.168.1.1 and change the settings.

You might try this link to see if it is a solution for you: http://barugon.wordpress.com/2009/03/08/constantly-resetting-linksys-wrt54g-router/

I have to assume that you have turned on WiFi in your iPhone. If you have made your router's SSID hidden, please un-hide and try again.

You are looking for a Wireless Security key, this is not the Linksys password for the admin portion, look on the bottom it maybe there if not changed. Else goto the router home page and check it there something like 192.168.0.1. or 192.168.1.1 etc etc

did your run the router software on the new computer? Give it a try.

Dont have it? Go to www.linksys.com/support

Go into the admin settings of your router using http://192.168.0.1. Look up the wireless section and make sure that wep encryption is set to 64 bit.

1).turn off the modem and the router.disconnected from the Internet ISP. and the computer.
2).disconnect the router from the modem.
3).turn on the modem.
4).connect the router to the modem, and turn on the router.
5).connected the computer to the router, adn turn on the computer.
if you are using wireless, waited about 5 minutes before turning it on.
goto control panel>>wireless network, and configure the wireless seting.