Windows 7 Ultimate 64-bit.
I'm behind Linksys WRT45G. For some reason, I need to run ipconfig /release and ipconfig /renew on every reboot to get the internet working.
This could be multiple things.
1) Your NIC is not properly cycling the connection on reboot. Update your network card drives, and chipset drivers (they may be influencing this too).
2) Your router may not be properly running it's DHCP server. Seek a firmware update from Linksys.
3) Your router firmware is bunk (likely). Consider upgrading to dd-wrt 3rd party firmware, or tomato 3rd party firmware. Tomato is really well done, and I know supports your model. DD-WRT supports a much wider array of devices. Both firmwares are great, but if you are new to using 3rd party firmware I recommend working with DD-WRT. Don't worry, if you mess it up, linksys devices (such as that one) have many failsafes to enable you to re-upload a working firmware, even if you broke a firmware update.
4) Your cable is damaged and connectivity is limited.
5) You have some sort of bogus rogue anti-virus software that is overprotective, such as Norton.
If I'm reading this right, your DHCP lease expires at the same time it is obtained. I would look at the DHCP settings of your router.
I can tell you that I have the same issue with my Linksys WRV200 wireless router. I usually power cycle the router to fix the problem. Note that I'm not using the DHCP service on the Linksys (I'm using an internal DHCP server) so the problem doesn't appear to be with the DHCP service on the Linksys router but with the router itself.
Have you tried upgrading the firmware on the router?