police
Like the rest of Germany, the whole area of Eastern Germany is a very safe place to travel. The laws are strictly enforced, the police are easily approachable and very likely to help. In the course of the Football World Championship 2006 the police force of the major cities received enough training to even deal with foreigners in English. In smaller cities you will find the need to talk German.The emergency phone number is 112 for every type of emergency.
beggars and pickpockets
While the German social systems allow everyone a reasonably decent living from social welfare, bigger cities such as Berlin or Hamburg have their share of beggars and homeless on the streets. Those people are harmless and closely watched by the police.Especially in the vicinity of train stations or in growded pedestrian zones you should be aware of pickpockets. Again, there is nothing extraordinary, and common traveller's sense will suffice.
non-white visitors
While large cities such as Berlin and to a smaller extent Leipzig and Dresden have become multi-ethnical and have large immigrant communities living there, in smaller cities and in the countryside, non-white visitors might sometimes attract wary looks. In some places there are right-wing Neo-Nazis, which may look for trouble, especially when drunk in the evenings. Act with common sense, and in doubt stick to bigger crowds of 'regular' people. Don't be afraid of calling the police when feeling uneasy.
homosexual travellers
Germany is very open-minded towards homosexuals. Nowadays Germany even has a gay foreign minister and the city of Berlin is ruled by a gay mayor. Current law is in a process of change and gay/lesbian couples can even have same-sex marriages with a number of fiscal rights attributed to that.
Overt display of homosexuality is no problem in big cities such as Berlin, as they have their own gay/lesbian communities. In the countryside and in conservative areas displays of homosexuality can attract wary looks, and with Neo-Nazis around may result in insults or worse. You are better off, applying common sense.