Kazakh and Belarusian banks open cards for Russians with no questions asked. There are services that do it online for you. You only need a passport (the international kind).
In Armenia, at least when I visited in December, I was told that "if you want an Armenian card, it can be done in 2 days for 25000 rubles that you put in your account". Not sure which bank it was.
I also know people who opened cards in Indonesia and Turkey.