Hi I've got the same Linksys Wireles ADSL - WAG54G.
If you've replaced the old one and the same thing happens with the new one then its not the router but the connection and or connection settings.
So firstly, how is the router connected to the phone line? Do you use extension cables? If so this may be part of the problem as I discovered when I had similar problems.
It turns out that the longer the telephone extension cable the weaker the ADSL signal gets so I had to go on eBay and buy a longer direct ADSL conetion cable (sorry I can't remember what they are called but you know the cable with sqaure shaped head that comes with the router and your ADSL) between the router and the main telephone line entry point into the house.
Also check your configuration settings on the routers connection address.
http://192.168.1.1
and make sure they are setup correctly for you ISP's ADSL network setup.
Also try and get the latest firmware for the router for your location. Linksys are a bit dodgy with thier firmware because it can be a nightmare trying to figure out which Annex version firmware you are meant to download.
Sadly I can't remember the link to the Linksys ftp site but you can google "Linksys WAG54G firmware download ftp site" and you'll eventually get there even if indirectly through other furoms. This was how I managed to update mine recently. The reason is Linksys don't publish their ftp site for reasons only known to themselves.
You may also need to need to connect to the router using a 10/100m ethernet connection, because I'm having problems connection to the configuration http: with a wireless connection now that I've protected my network, though everything else, i.e. the internet works wine.
Hopefully this will work for you