The latest Xbox 360 dashboard update has advertising built in, but fortunately you can simply setup your DNS to block the domain the ads are served from. Following a helpful post on Reddit I assisted my cousin with blocking the ads on his 360. Doesn't fix all the problems with the new dashboard, but it's a start. There are two ways you could do this, either using OpenDNS or configuring your router to block the ad domain.
In my case I configured the router to block access to the domain rad.msn.com which is where the dashboard ads are coming from. How this is done varies based on router model and firmware, but I think any decent router will have an option to block access to a domain. According to the Reddit thread this doesn't work properly on some routers. The alternative is to setup OpenDNS, if you aren't using that already.
The OpenDNS method requires changing your 360 to use the OpenDNS servers by setting the network settings on the 360 to manual and entering the OpenDNS primary and secondary IP addresses, which are 18.104.22.168 and 22.214.171.124. Then you'll need to sign up for an OpenDNS account if you don't already have one, create a network for your IP address, make sure domain typo correction is enabled and then filter the rad.msn.com domain. The Reddit thread has more detailed step by step instructions. Note that if your IP address is dynamic you'll want to setup software on your computer to update your OpenDNS settings as it changes with the OpenDNS Updater or something similar.
Whichever method you follow, you may need to clear the hard drive cache on the Xbox to remove the ads that have already been downloaded. This will remove all the title updates for all your games, so you'll need to redownload all of those.
This does not block all the annoying graphics on the new Xbox 360 dashboard, just the advertisements on the small corner tiles. Better than nothing, I suppose.