Unlike England, Germany back then was no centralized country. Generally speaking, in a centralized country, the dialect spoken in the capital is regarded as the most prestigious one and will over time influence other speakers. Modern French is essentially the dialect spoken in Paris plus a few innovations. This means that a modern guy from France would probably be understood if he travelled back to Paris 500 years ago but would have a hard time if he visited a different city like Toulouse or Marseilles. Similarly, the the dialect spoken in London had a great impact on modern English. If you, as an American, travelled to London, you'd be fine but things would be different in southern Scotland or Northumberland.

In Germany, the situation was quite different. There was no unified nation and no capital whose dialect could be regarded as most prestigious. Modern German is essentially a mix of different languages and dialects, created by Martin Luther who first translated the bible into German. He tried to appeal to all Germans and therefore refrained from relying on a single dialect too much.

To put it in a nutshell, Standard German nowadays is a mixture of Eastern German grammar which was based on medieval High, that is Southern, German grammar while the pronunciation is almost entirely Low German. Low German itself is a very different language, more close related to Dutch and English than Standard German.

To sum it up, you would most likely have a hard time adapting to the German spoken back then.