Biggest problem is always buy-in. If any team, or key individuals haven't bought in (project management, QA, development, etc) then failure is almost assured.
Another related problem is actually making everyone involved aware of what scrum actual is and what is not.
I've seen environments where project management has actually taken this as a ticket to come directly to developers with changes and expect it to get done tomorrow, since we are using the great new process. Anyone that has been in this situation or in other failed attempts at implementing Scrum and have a bitter taste in their mouths. These people sometimes will try to de-rail the project as well.
Another problem I've seen is stand up meetings. You will always get the guy that wants to sit down during a stand meeting.... "I've got a bad back" or whatever. It always seems to be the same guy that has no idea what the objective is behind the standup, and will not shut up about politics or what he did that weekend. Stand up meetings, I have found, are the key to effective communication. It is important not to let anyone poison these meetings.