Hi David,

Welcome to WebmasterWorld!

If you change the preferred domain name setting in Google Webmasters Tools to the version without the www, Google will know that it is the same site and change the URLs in the following weeks. But you should know that this option in Webmaster Tools is only a hint to Google and if many external links point to the www version of your domain, the algorithm may still decide to use that version in the SERPs instead.

A proper 301 redirect from the www version to the non-www URL is a good idea to be sure that all new external links you acquire point to the non-www version. But you may loose some pagerank with this as you can read in this recent thread [webmasterworld.com].

As much as I hate those redudant characters at the start of a url the general public understand www as meaning that the character string is the address of a web site. Unless you have some major reason, beyond the aesthetics of the matter, I would recommend sticking with what Joe Public is comfortable with.

Let the site run with www URLs. Doing that makes things much easier to diagnose canonicalisation problems using [site:www.example.com] and [site:example.com -inurl:www] searches.

Ensure you have the site-wide 301 redirect installed. Do you ever go to 'www.google.com'? No. I expect you type just 'google.com' - look what happens when you do that.

You can still advertise your site as 'example.com' and the redirect will get visitors to the correct URL should they choose to type just 'example.com' into their browser.

The danger comes in allowing both www and non-www URLs to both return '200 OK', rather than one of them correctly sending a 301 redirect OR once the redirect is in place, having any internal navigation links that when clicked send you through a redirect instead of directly to the correct content page.

In Google WebmasterTools, make sure your register and verify both www.example.com and example.com and that you look at the 'internal links' and 'links to your site' reports for both. It can be very revealing.

you may loose some pagerank with this as you can read in this recent thread [webmasterworld.com]

it is important to note that MC's comment was specifically in response to EE's question that began:

Let's say you move from one domain to another ...

this doesn't precisely apply to this situation.

you might want to revisit the www vs non-www home page discussion from a few months ago:
To www or not www the domain*com [webmasterworld.com]