If you want to cross South America or Africa on the North-South or East-West axes, you will have to cross plenty of countries. This makes it very difficult to find nonstop trains, as train travel between countries is not always obvious.
Regarding Europe the situation is similar. Within a vast majority of countries you have decent rail networks. However, when it comes to traveling between countries, the situation is a bit more complicated.
Nevertheless you have a some interesting options in Europe too. Okay, they are probably not covering the whole North-South or East-West ranges. On the other hand, the bumpy shape of Europe makes this difficult anyway.
- From West to East: There is a daily direct train from Amsterdam to
- From North to South: There are direct trains from St.
Petersburg to the Black Sea (Sevastopol and Sotchi). These run two or
three times a week. If you are ready to accept a train change in St. Petersburg, this route can even be extended beyond the arctic circle.
I know that this is a poor comparison with the Trans-Siberian, but not too bad for European standards ;-)