A less known feature of cyrus imapd is the squatter index daemon. It has to be activated in the /etc/cyrus.conf:
squatter cmd="squatter -s -r *" period=180
to index new messages in all mailboxes every 180 Minutes.
The new Thunderbird creates another index locally. But this makes Thunderbird work very very slow the first days if you have some huge mailboxes. And this happens on every computer you use Thunderbird 3.0. So you might want to deactivate it. You can:
Edit -> Preferences -> Advanced -> General: uncheck "Enable Global Search and Indexer"
I really did not test which search is faster. I use (cyrus) imap built in search for long time now and it takes only seconds to search some GB and some hundred thousand Mails - Full text!
It's the same search field in Thunderbird you have to use ...