A kind of gathering place for Russian independent animators is the website koms.ru. Some of the best artists are Andrey Bakhurin, aka. Scarydoll (my favourite of his films is the beautiful Los Dias Sin Dias, made in 2003) and Nikolay Belov, aka. kol-belov, who has pioneered a very singular style that has been imitated by others. His typical films are wordless visual metaphors in which shapes are constantly morphing (a technique that suits Flash like a glove). They tend to be very dark and often hard to understand. ATM (aka. Bankomat, made in 2006) is one of his films where the message is quite obvious. I've felt that Belov's films have been going down in quality for the last few years - these days, they tend to feel rushed or not genuine.
Those films that I linked to are both wordless. I'm now going to list some recently-released films with English subtitles.
(if you don't see subtitles, click on the triangle at the bottom right of the Youtube video, and make sure that CC is turned on)
Mr. Freeman, parts 0, 64, 63 and 3 (in that order). This is a recent series of philosophical cartoons, the first of which was released 2 months ago. I could not find out who the director is, or who did the English translation, or much info about them at all. They've been watched hundreds of thousands of times, so I guess something in them speaks closely to many Russians.
"Diglator" is a film based on a short story by the Polish science fiction writor Stanisław Lem. It was directed this year by Aleksandr Menshikov.
"Now, Listen!" is a great series of funny fairy tales made by Andrey Bakhurin, who I mentioned earlier. Released in 2008, the first eight of them have English subtitles: