See http://askubuntu.com/questions/1916/how-can-i-access-system-mail-in-var-mail-via-thunderbird.

Create the account
Thunderbird -> Edit -> Account settings -> Account Action -> Add Another Account -> Unix Mailspool (Movemail)
Your Name: adr (mailspool)
Email Address: adr@localhost
Browse your mails
Open Thunderbird then press Get Messages button; you can now see the messages from /var/mail/adr.

See your mails in console
cat /var/mail/adr

Redirect cron mails to you
When running cron jobs with other user (e.g. root) configure MAILTO****@adr-desktop to received a cron-status.

command for sending a mail to adr@localhost
/usr/bin/printf "%b" "***** your mail body here *****" | /usr/bin/mail -s "** your mail subject here **" adr@adr-desktop