The first "three" Call of Duty's (CoD, CoD:UO, CoD 2) all had Russian campaigns (along with US and UK) most of which were centered around Stalingrad and/or the attack on Berlin and raising the flag on the Reichstag. The first game recreates Enemy at the Gates, where you're issued a clip of ammo and have to pick up an actual rifle off a dead comrade. Then it also does some counter-sniper stuff. The "big moment" of the first game is defending the siege of Pavlov's House.
UO sends you into Kursk and has a huge, for the time, tank battle between you and the Germans tearing up a bunch of Russian villages.
2 does Stalingrad again but has essentially one map where you experience offensives and counter-offensives from different parts of it. (ala you capture a factory from the Germans, and then have to defend it from their counter-attack, and then you go on your own counter-attack against an area you fought in at the start of the campaign but had since been captured)
Arguably the best moments of all three are set in the Soviet campaign. Other than UO's take on the Bulge.
Finest Hour for PS2/Xbox/GCN is pretty much garbage but most of it is set with the Soviets and it does the Enemy at the Gates, etc. rip offs. 2 on PS2/Xbox/GCN follows the American 1st, was Treyarch's first game and was probably the only good console exclusive Call of Duty.
The idea of 3 was to follow along a single European front, albeit among the different allied forces. World at War's non-US campaign is Stalingrad and Berlin again. But I didn't think they did Stalingrad as well as IW did. I seem to remember Berlin being more fun though.
The ultimate Eastern Front FPS games are the Red Orchestra series. The first two Sniper Elite games take place during the Soviet invasion of Berlin but you're an American OSS agent undercover as as German sniper. I want to say that in the first one you actually fight for the Nazi's at one point to save certain members from the incoming Soviet forces. Though you wind up fighting for pretty much everyone (it has a bit of a faction system that doles out the missions) it's one of the few games where I remember you actually fighting for the Nazi's in the campaign mode. And they aren't like good Nazi's, but ones the U.S. wants for our own nefarious purposes.