In the BBC radio 4 series The Wild East, the history of Russia after the Mongolian occupation is described as if all the bad things happend subsequently had been caused by the legacy of that era. Isn't there any positive legacy of the Mongolian occupation in Russia ? For example, didn't the newly opened up trade route to the east benefit the economy of Russia ? Wasn't the cultural influence from the east one of the appeals of the Russian music to the West European audiences in the early 20th century ?
Interesting question and highly creative conjectures but ultimately the answer is no. The trade routes didn't play much of a role: Russian trade remained oriented on Western Europe; as for the great Russian music, it was the product of the 19-20th centuries and followed and developed, once again, Western patterns.
So, to sum up: I can't think of any positive influence the Mongol overlordship had (it's a more accurate term then occupation, in this context). The negative legacies are legion.
EDIT: I was asked to provide references. Right now on the fly I found this chapter (in Russian) from a recent (2001) book called "Ten centuries of Russian mentality". I'll quote a bit in the original:
For lack of time I'll have to make do with a slightly-edited Google Translate translation:
my guess is that cultural transfer from mongols to russians was huge.
no that they teach russians to build ships, calculate trade and play flute, but consider this:
sophisticated 'honour' system between lords distinguish russian from european mentality until now. and allow russians to survive many turbulent periods. mongol operation of masses and armies was more eye-eye, body-body contact and feeling to feeling, rather than western word-by-word, paper-by-paper approach. and russians seem learned that from mongols to my vision. russians will follow the leader that will breathe and blink at a proper rate rather than the one who speaks right words about freedom.
even these days this approach allows putin to consolidate forces around kremlin. this is definetly positive if existance of russia is considered positive.
then russians learned about asia from mongols, which later allowed to extend small country to the pacific, to persia and to china borders. is that positive?
yes the idea that russia those days was not occupied but rather like a state within US is accepted by many russians. then invasions should be called local suppression.
above someone mention religion-tolerance. yes, seems to me, russian culture in this area is also seem to be mongol-like. russian will not follow religious rituals in detail but would try to achieve God's(or ancestor or spirit) support in their activites. Russians for centuries would base their strategic decisions on their religion experience. Last czar planned WW1 activities based on "advices from god" he received while praying or from prophets (like rasputin).
just like mongols russians will support or leave alone those who pray the 'right god', and cast away or kill those who pray to a 'wrong gods'. this is quite positive, in some sense. as western style of religion-tolerance is to leave alone anybody who claims to do praying.
of course mongol unwritten 'honor system' or 'the way to communicate and cooperate efficiently in sophisticated manner across large areas in long term over generations without breaking apart in small fractions' is what holds russia together until now. this is most positive outcome. I cannot see how russia would survive so close to burning europe without that "we are huge together" mentality of mongols.
Some historians argue that the Mongol invasion had a very large positive effect.
At that time Russia had 2 enemies:
Russia couldn't fight both of them and had to decide, which of them was lesser evil - should it become part of the West, or part of the Mongol empire?
If Russia chose to become part of the Western world, then it had to convert to Catholicism. When Western powers conquered a certain territory they usually required the population to adopt their religion.
Had Russia been converted to Catholicism, it would later suffer from the wars between Catholics and Protestants. Some historians claim that these wars eliminated from two thirds (Germany) to 75 % (Czech republic) of the population.
The Mongols, on the other side, didn't care what religion the Russians practiced. They only cared about the taxes. The Mongol occupation prevented Russia from participating in bloody religious wars and the resulting casualties.
As far as I remember one of the politicians, who contributed to Russia becoming part of the Mongo empire, was awarded the status of a saint for this reason.
Long story short: The symbiosis with the Mongols helped the Russians survive.
I would recommend reading Lev Gumilëv works if you'd like to look onto positive sides of Mongol (or Tartar-Mongol) occupation, starting with this assay [in Russian] and its main source (1). One of his points is that Mongols were allies in struggle against Teuton and Livonian orders.
As to neutral impact, one should first of all look at the numerous loanwords from Turkic languages, such as e.g. 'Kreml'
1) Гумилёв Л.Н. От Руси к России. - М.: Прогресс.
Also Karamzin, who was mentioned above is of Turkic origin (name coming from Turkic 'Kara Murza', Black Prince).