China is not one continuous dynasty. No dynasty lasted as long as the Romans.
Reading about the Byzantine Empire just now[1], I'd say 476 to 867 was one "dynasty", because in 867 marked the beginning of a Macedonian dynasty, which lasted until 1025. This would break down the Roman empire into dynasties of length not unlike Chinese ones. How do you feel about that?
The Macedonian Dynasty in the Byzantine isn't quite comparable to Chinese dynasties. In China, a dynasty change, AFAIK, meant that one group overthrew another group and established their own nation, with its own laws and government structures. In Byzantine the change to Macedonian Dynasty happened as a result of internal politics (+ assassination). The nation remained the same; just the guy in charge changed.
Not that it isn't impressive or anything, but that isn't a millennium!