I use Outlook myself, and have never had no problems with it, I haven't used Eudora or Thunderbird, recently I have moved to Outlook 2007 and yes it is Microsoft but the functionality is amazing, couple with the p910i phone, it works brilliantly, my only complaint is that it is a little bit resource hungry.
Incidently if you are using gmail, then follow the link Joxley has given and then you have to set up rules, otherwise email you have sent using the web version will go into the inbox and not into your sent items. There is a handy tutorial over at Computer talk.