You might travel all over the western part of continental Europe without having to show your passport. However at the Danish-German border you will be checked no matter what. In order to keep the bureaucrats happy, bring your passport. Also, do not forget to apply for a visa, if necessary. Contact the Danish embassy in your home country.
Generally, visitors from the US and Europe as well as Australia and New Zeeland do not require a visa to enter Denmark. For more detailed information please see http://travel.state.gov/denmark.html
