I use a Linksys router, and have had it going for almost a year. Since getting it, and having a network of about 5 computers, I have always kept the same IP address. Even when I took my computer off of this network and brought it back, it gave me the same IP again. The way this works, is that when you are assigned an IP address, it has an expiration, instead of renewing every time. You can see this information in a Dos window by putting in "ipconfig /all"
When the lease expires, the router will more than likely just assign your computer the exact same IP.
However, if you really want to, you can manually set up your networking properties manually. You just need to use the same IP range as what the router would give you.
Example:
IP: 192.168.1.50
Gateway: 192.168.1.1
Subnet Mask: 255.255.255.0
Also, Linksys routers have an option to set the starting DHCP address. I don't know if it is required, but I would recommend manually setting your address to a number lower than where the router will start assigning on its own.
Hope this helps!