I imagine that due to suppression of religion traditional ceremonies in churches stopped. Did they still look pretty much the same minus any mention of God, or did the Soviets try to create something completely new?
This is very insteresting question.
Early Soviet Union tried to create something new in every aspect of human life - family, army, language, sex... you name it.
Marriage wasn't an exception - in early Soviet days marriage wasn't even required to be official. People who wanted an official ceremony in 20-30s (which often was called 'Red Wedding', yes, same name as in Game of Thrones) created an informal event with L'Internationale as background music and communist speakers as masters of ceremonies. Of course, visits to church were out of the question.
Otherwise if people didn't wanted any official ceremonies they can even not bother to register their marriage officially.
Long story short, this freedom lead to problems with demographics and Soviets started to turn marriage back on previous 'traditionalist' tracks. But marriage ceremonies per se were not formalized up until 1964 when Soviet Council of Ministers published a law about 'civil ceremonies'.
In many cases there was no ceremony at all, the newlyweds merely obtained a marriage license. Traditional wedding ceremonies were also sometimes carried out in private.
I can provide a specific example of a Soviet wedding: John Scott, an American living in the pre-war Soviet Union, married a Russian woman he had met in the factory where he worked.
In his autobiography Behind the Urals, he describes his 'wedding' as being a brief and unimportant affair. He and his new bride went to the local Magnitogorsk soviet, obtained their marriage license, and were married immediately, returning home that day with no fanfare.