Since very long, we have known that Amazon will be releasing an online Kindle books reader, called Kindle for the Web. Now, it is live and available for quick sample viewing.
A book's product page has a button that says 'Read first chapter FREE' instead of the 'Get Sample' button. Samples are still available through the Store on a Kindle device at the time of writing this. But it appears that the sample-downloading-to-PC is gone.
Kindle for the Web is in beta for now, so there might be rough edges on it. But you can give it a spin by browsing to any book (that is available in your location) on the Amazon site and clicking the button for yourself.
More more details, you can check out Amazon's help page over here.
Quick Edit: It seems that the paperback version page has these buttons in the right pane, while the Kindle version has the 'Send sample now' button. It's not gone yet.