Okay, first off there's a small but important difference between Gyros and Doner (I'll just say Doner from now on since the Umlauts don't work here):
Gyros is, usually, made from pork. D?ner is, usually, made from either lamb/mutton or chicken/turkey. Some Doners are even beef-based and/or contain minced meat. Gyros is, where I live, ALWAYS made from pork and never from minced meat.
The typical Gyros spices are garlic, thyme, oregano, salt and pepper. Sometimes (but definitely not necessarily) it also features cumin, majoram and coriander. However, these last ones are optional.
The typical Doner marinade contains garlic, thyme, oregano, salt, pepper, chilli, paprika, onion, tomato and parsley.
So, they really are two separate beasts especially since a Gyros is served with pretty thick Tzaziki with cucumber bits (easy to make!) and coleslaw whereas a D?ner gets served with mixed salad and (usually) a red hot sauce and/or spice ("Pul Biber" or "Harissa", depending on where you go) as well as a thin joghurt sauce.
If you would like to reconstruct the "german" Gyros taste from scratch you should start with pork and just the bare neccesities (ie skip the cumin and coriander, feel free to add majoram) - DO NOT ADD CHILLI TO GYROS!