Always when I start Chrome I get a prompt for my keyring password. It seems like Chrome doesn't save anything in it. So, why does Chrome ask for the passoword?
Unfortunately, older versions of Chrome must unlock the keyring to be able to check if you've saved passwords for a particular site. This has been fixed in the current development version; Chrome will now only prompt to unlock your keyring if it actually needs to retrieve a password from it. If you don't want to use an unstable version of Chrome, you can work around the issue by passing the
--password-store=basic option so that Chrome uses it's own password store rather than attempting to use the GNOME keyring.
For more information, see bug #85285 in the Chromium bug tracker.