This is a controversial issue indeed. There exists no concensus among historians regarding the role of the Mongols in Russian history, even when you consider the fact that the Mongols wielded influence over the Russians for almost 250 years. In Russian circles, it is the tradition to repudiate and downplay the Mongol influence on Russian history. This was the ruling consensus until the 1920s when especially Russian émigré intellectuals with their so-called Eurasian school of thought. As the name suggests, the Eurasianists proclaimed that Russia history has its ties predominantly in Asia (i.e., from the Mongols.) and that the "Mongol yoke" as the Russians like to call the Mongol dominion brought positive change to Russia. Their theories are doubtful, because they far more biased than the remaining majority of historians (they're often associated with Soviet historians and many Eurasianists made careers in the Soviet secret police).
Consequently, I would say that it's important bearing this in mind when analyzing Mongol impact on Russian society.
No one can deny that the Mongol invasion from 1237-1240 brought wholesale destruction to Russia. On their way through Russia, the Mongols brought unprecedented carnage and devastation to a society advancing far beyond what the Mongols had ever achieved. Nevertheless, the Russian society and its population were completely exterminated. To quote a papal legate and archbishop of the time, Giovanni da Plano Carpini:
"...they went against Russia and enacted a great massacre in the Russian land, they destroyed towns and fortresses and killed people, they besieged Kiev which had been the capital of Russia, and after a long siege they took it and killed the inhabitants of the city; for this reason, when we passed through that land, we found lying in the field countless heads of bones of dead people; for this city had been extremely large and very populous, whereas now it has been reduced to nothing; barely two hundred houses stand there, and those people are held in the harshest slavery." [Riasanovsky, Nicholas, A History of Russia, OUP, 2000, p. 72].
This very aptly describes the magnitude of the destruction waged on Russia and its people. And this destruction, it is estimated, might very likely have set back the development of Russia by 150-200 years. Some even contend that had the Mongols not invaded and occupied the country, it might very likely have been able to take part in events such a the Renaissance and the Reformation. I agree with Rabid Pop Tart that the Mongol yoke likely spurred some of the features of the Russian national character. I think it generally induced some harshness and tenacity into the "public spirit", and who knows how the Russians would have handled WWII without the experiences of their forefathers? Whether all this is positive is, of course, subject to discussion, and it will hardly ever be substatiated further to what extent the Mongols shaped the Russian national character.
One of the substantial, "positive" Mongol contributions to Russian society and culture is its influence on the Russian language. E.g., the words деньги (money) and ярлык (label, tag) derive from Mongolian. The Mongols generally had a profound impact on the Russian language and culture.
Moreover, it can argued that the Mongols also had an impact on later Russian military tactics and evolution of the Russian military, especially the cavalry. However, people often mix up the fact that Russian appanage warfare based on ordinary foot soldiers was a Mongol trait. Instead, this derives from Kievan Russia. But again, the "positive" aspects of the Mongol yoke are limited.
Lastly, it is importatant to appraise the fundamental differences between Russians and Mongols. While Mongols were primitve nomads, excelling only in warfare, the Russians compared, were highly developed agrarians with a set of laws often considered to be ahead of its time. Some, especially the Eurasianists, like to compare the influence of the Mongols on Russia with the Arab influence of Europe - this, nevertheless, is hardly true at all. When you look at the other Mongol states - its dynasty in China which lasted roughly 100 years, the Mongol dynasty in Persia which lasted from 1256 to 1344, and the Mongol Central Asiatic state lasting only around 128 years. The common denominators of all these states are corruption, instability, and inferior statecraft And in Russia, they never even established their own dynasty, instead they acted as overlords. Muscovy was not a state based on a Mongolian model, rather a combination of Byzantine and Kievan statecraft.
