A recent theory has it that Sumerian is to be considered closely related to Uralic. It is proposed by Simo Parpola, an assyriologist:
This is not the first time that such an idea is proposed: other people have proposed that Sumerian was closely related to Hungarian. The difference in that case is that the theory is proposed by a scientist, well-known for his expertise on Assyrian. http://en.wikipedia.org/wiki/Simo_Parpola
So what is to be thought of that "new" theory?
To start with:
On the whole, issues of genetic relationships are twofold:
Q1: why should Sumerian be considered related to Uralic at all?
Q2: why should Sumerian be considered more related to Uralic than to other families?
Just to take one example, the paper states that Sumerian may have had labiovelars like *kw or *gw and alternations like b/g may originate in such phonemes. Why not after all?
But this raises a number of issues:
Q1: is this the only way to explain that graphic alternation? => I'm afraid not.
Q2: now, assuming that the idea is correct, then we can see that Sumerian igi, ibi "eye", possibly from *igwi according to that line of reasoning, is structurally the same as Proto-Indo-European *Hekw "eye". So we can just as well promote the idea that Sumerian has ties with PIE as well.
Q3: note that Uralic Vogul-Mansi has a verb ankw- "to see", suggesting that ultimately all these languages: Sumerian, PIE, Uralic may be related. It's not unusual that Uralic has a kind of infixal -n-.
=> in that case the correct framework to investigate Sumerian's relationships is Nostratic.
It is indeed probable that Sumerian is to be included in the Nostratic superfamily. The issue is then: in which place and which subfamily?
The theory proposed has it that Sumerian is closely related to Uralic at a very recent dating. In fact this dating, about 5000 years ago, would make Sumerian a Uralic language, if Sumerian is indeed related to any single one Uralic language on that basis. Relationship at such a recent dating means *inclusion* within Uralic perimeter not relationship with Uralic.
=> this seems impossible at first glimpse.
The common lore about Proto-Uralic is that the original Uralic community originates in a Paleolithic culture. This is a point that Mr. Parpola does not seem to know, assyriology being his core expertise. The claim that there are "common words for cereals, sowing and harvesting, domestic animals, wheeled vehicles, and the harness of draught animals" between Uralic and Sumerian is not far from nonsense. From the Uralic point of view, such words can only be either loanwords, wanderworts or innovations. Most of them happen to be of PIE or IE origin.
A typical case for example is:
Sumerian urudu "copper"
PIE *Hreudh- "red"
Finnish rauta "ore"
A clear wanderwort of probable PIE origin.
Similarly (eastern dialectal) PIE *yewos "grain" => ancient loanword PU *yuw, Finnish jyvä. One more wanderwort. Etc.
It must be borne in mind that Finnish and more generally Finno-Baltic is not especially representative of (Proto-)Uralic phonology. These languages have a high frequency of vowels in words and a rather simple syllabic structure of the CvCv type. This feature makes it easy to compare Finnish (or Finno-Baltic) with about anything else with a similar structure.
For example Finnish can be easily compared with Tamil or Dravidian. Most of these comparisons crumble when they are put back in the Uralic perspective.
In that respect it is especially troublesome that Sumerian seems to have strong ties with Finno-Baltic and very poor ties with Altaic. This is a telltale indication that the conclusion of the theory is an artefact of the comparanda taken into account.
In all cases the discussion of the relationships between Uralic and Sumerian only makes sense, when putative close relatives of Uralic like Yukaghir are involved. What does that theory teach us on the relationship between Uralic and Yukaghir?
For example, as regards the word "brain": 1. Sumerian ugu, "top of the head," "brain."
2. From the Uralic PoV the reconstruction as *ajkwo "brain, top of the head" is completely unacceptable and procrustean. The usual reconstruction for Finno-Ugric is *ajN "temple, skull", secondarily "brain, marrow". Samoyedic is *oN-pu "temple". It is possible that Finno-Ugric *ajN is a kind of compound < *aj-oN- or the like. What is to be compared is *oN- with ugu. Apparently Sumerian is ugu not **ug~u. So it does not work on phonetic grounds. => Status: highly dubious.
In all cases it makes no sense to compare Sumerian with Uralic when Uralic data are not properly dealt with.
there is no Volgaic word kurok "mountain". In Mordvin kurok means "quick, soon" = Russian skoro. In all cases, kurok, if it exists at all and is not a recent borrowing of Russian gora, can be related to some reflex of PIE *gwer- mountain, which makes sense as a cognate of Sumerian kur "mountain". And we are back to the starting point that Sumerian has fairly strong ties with PIE, which may be much clearer and more workable than the alleged ties with Uralic.
As a preliminary diagnosis, I would classify that theory as amateurish and underinformed. I'm afraid there is no "constructive criticism" possible.