The first step is to enable non-secure imap. In order to do this, /etc/cyrus.conf should have:
imap cmd="imapd" listen="imap" prefork=5
instead of
#imap cmd="imapd" listen="imap" prefork=5
After this, cyrus-imapd should be restarted with:
/sbin/service cyrus-imapd restart
Now we can connect to non-secure imap service:
cyradm --user cyrus --server localhost --auth plain
and add a user:
createmailbox user.logon
We are done.
The last thing is to disable regular non-secure imap again in /etc/cyrus.conf:
#imap cmd="imapd" listen="imap" prefork=5