As a resident, I can tell you it varies depending on what part of Concord you live in. Contacting the local police department or using the "Local Info" tab is a great start.
In my particular zip code, it is minimal... mostly older established neighborhoods.
Areas of Monument BLVD can be pretty bad as well as Solano.
Towards the Clayton border, as well as the northern section of Ygnacio Valley are really nice, and safe.