Dont walk alone at night, too many foreigners have been violenty attacked since the begining of 2005.
(like myself) Most of the poorer expats that dont have security at theyr home or body guards, have been agressed violently at least once while in Mongolia.
And if you go to Markets, hide your valuables in a money belt undwer your clothes, and avoid wearing a handbag.
Pick pocketing and Crime is a serius problem here, but you can avoid those if you prepare yourself.
You can rent a safe on a daily basis in some Banks.