I spent a couple of hours googling the Internet to find support for my observations.
I found an interesting book called Tense and Aspect in the Languages of Europe on Google Books that answered my question .
Here are some excerpts from it (highlights are mine):
One, which is relatively unproblematic to ascertain, is the absence of inflectional futures, which holds for all Germanic and Finno-Ugrian languages and for the majority of the Slavic ones; as well as for some more peripheral parts of Romance and for the non-Slavic languages in the Balkans. Another manifestation of a slightly more elusive kind is the tendency not to distinguish present and future time reference in any systematic way at all, be it inflectionally or periphrastically.
It also notes that different grammatical ways of conveying futurity found in the Slavic languages are a relatively recent invention:
Similarly, there seems to be no evidence of any grammaticalized future markings in Finno-Ugrian. Furthermore, the area may also originally included Slavic. The different FTR devices found in the Slavic languages are, like the Germanic and Finno-Ugrian ones, relatively recent, with one exception: the use of the Perfective Present for future time reference, which, according to standard descriptions, goes back to Old Church Slavonic. However, there is good reason to doubt that this was a trait of Common Slavic. It is not in general found in the modern South Slavic languages, and there is some evidence that the link between the Perfective Present and future time reference was at least not fully developed in Old Church Slavonic.
I'd like to expand on my above answer as there seems to be a lot of misunderstanding and discontent in the comments about future tense in Russian.
I'd also like to mention that I'm not trying to impose my view on anyone who prefers to disagree with it. It is indeed a very debatable question that can be looked at from different angles. If you'd like to read more about what should be considered tense, I think there are great posts on ELU Whose tense is it, anyway? and When did periphrastic tenses stop being tenses?. The things discussed there can be applied to Russian as well.
I was asked in the comments to add more scientific sources to my answer so I will try to do it below.
Here's a nice write-up on the topic by Cyril Babaev:
The present tense had not quite simple usage. Besides its direct meaning as the present action, it could be meaning also future actions, because there was no simple future in Common Slavic. Usually Indo-European languages have some suffixes to mark the future tense: -s- is one of the most frequent of them, existing in Baltic and Hellenic languages, -f- is used in some Celtic tongues. But Slavs unified present and future, and any verb could be used in the forms pointed above and mean the future event.
He then goes on to say that in the absence of future tense, the aspects took on the role of expressing futurity:
Later Slavic languages, nevertheless, acquired a form of difference between future and present. The category of aspect (perfective and imperfective) possesses also the meaning of tense in Modern Slavic languages, and imperfective verbs which denote the action which is still taking place go for the real present tense (Russian daju - I am giving, from Common Slavic *dajo.), while perfective verbs denoting the final action usually mean the future (Russian dam - I will give, from Common Slavic *damï). Scientists came to a conclusion that there were not such a distinction in Common Slavic. Even Old Church Slavonic, which was spoken in 7-10 centuries AD, much later than the Common language, was just starting distinguishing aspects as present and future, and still mixed them.
As for the periphrastic "быть" type compound future, it arises from the lexical Common Slavic verb **byti* that means to be(come) and now it is used as an auxiliary verb to form future forms in most of the West and East Slavic languages. But "быть" is not the only verb that can be used to form compound future tense.
Currently, stati can be used as an alternative form of a future tense auxiliary in all East Slavic languages, as in the Russian example from a poem by Lermontov.
Stanu skazyvat’ ja skazku
begin(PRES.1SG) tell(INF) I story
“I will (begin to) tell a story” (Source: The Syntax of Compound Tenses in Slavic)
Plus, Balkan Slavic uses a descendant of the verb ‘to want’, instead of "to be". That's because
The nature of the future, necessarily uncertain and at varying distances ahead, means that the speaker may refer to future events with the modality either of probability (what the speaker expects to happen) or intent (what the speaker plans to make happen).
Thus, constructions like "Я намерен/хочу/должен делать что-либо" can all be used to express futurity.
I don't know about you, but I think it is a little unfair that we see "Я буду делать" as a special "future tense" form, but "Я стану делать" is not viewed as such. But I don't want to provoke another debate here.
Now the question is 'how come Russian grammars talk about future tense if it's not there?'
Well, I think that's because traditional grammars (also known as school grammar) are largely based on the principles of Latin grammar, not on current linguistic research in Russian. (This sentence was originally said about English, but I rephrased it for Russian, because I believe it gets the point across nicely).