I know that it hasn't been found (yet) and that we think his men diverted a river over it (source: Secret History of the Mongols).
What I want to know is does anybody have a plan or even the desire to find it? How likely is it to ever happen? Has anyone ever come close or thought they had found it?
Any help much appreciated.
Edit: Thank you all so much for your responses. Really helpful. Let's hope that it does get found one day.
While I'm reluctant to speculate on whether or not we will ever find Genghis Khan's burial site, I do know of there's at least one researcher trying to figure out it's location and I can give some insight into his research. Note: Most of this will be taken from a short presentation that I attended at UCSD, so I unfortunately cannot provide too many sources as far as his own evidence or conclusions go.
Albert Lin (1) is a researcher working with National Geographic and the University of California at San Diego. His primary tools have been satellite imaging to determine potential locations for a burial site based on geographic anomalies, as well as ground-penetrating radar in order to form images at locations. His search focuses on the Ikh Khorig (2) region of Mongolia, which is a historically holy area for Mongolians. He seems to have ignored some of the more outlandish myths, such as redirecting a river, in order to narrow his search to practical locations. Lin and a team investigated an anomaly on a sacred mountain, and due to a variety of factors Lin believes this to be the most likely spot of Genghis Khan's burial. In particular, that the site was found on a sacred mountain in a sacred region was compelling, and ground-penetrating radar found what looked like the foundation of a Mongolian tomb on the sacred side of the mountain. In addition, the team found what seemed to be shingles that were used around the time of Genghis Khan's death.
Unfortunately, further evidence will be difficult to gather because the Mongolian government is reluctant to allow any excavations in the region, so Lin cannot further verify if the site his team found is the tomb of Genghis Khan, or even a tomb at all. This seems to be one of the only modern attempts to locate the tomb, but I'm reluctant to comment on how likely it is that this is Genghis Khan's tomb. It certainly would be exciting if it proves to be so, and I think it is the most compelling thing right now. However, I think it would be unwise to jump to conclusions, and to Lin's credit, I think he has avoided doing exactly that, as neither his website (3) nor the National Geographic link mention the findings of his search.
Edit: I cannot figure out how to do links, so I'll just put them at the bottom.
1: http://www.nationalgeographic.com/explorers/bios/albert-lin/ 2: http://en.wikipedia.org/wiki/Ikh_Khorig 3: http://www.albertyuminlin.com/About.php
Simply put? It's almost certainly not going to be found. If it is found (and for all we know it could have been found already), we'll likely never identify who it is.
Mongol's at the time were buried without markings. Without a tomb or any indication, Genghis Khan's grave is just one of thousands. Considering that there's no consensus on how he died, any body found would be even harder to identify.
There's varying conflicting reports about how it was hidden (a river was diverted over it, or horses stampeding over it are the two biggest ones). But even without those, the odds of locating a specific, unmarked grave for a person who lived close to 800 years ago are nonexistent.
In 2004 they found his palace (which has been called a 'mausoleum' by several news sources including the bbc, causing a whole lot of confusion). There is some speculation that he might be buried nearby (because some texts say that officials travelled from there to the burial site to conduct rituals), but considering that conflicts with most other accounts, it seems questionable. Even if it is true, the only thing that narrows down is that it's within a half days ride of the site, which is still an extremely large area.
While it's one of archeology's great searches, there's just so much conflicting information that makes it very unlikely to ever be found. It might be under a river, or it might be under a plain, or a forest. It might be in the Burkhan Khaldun mountains, or it might be near Khentii Aimag. It might be a single man, unmarked, or it might be several men. There might be a baby camel in the grave, or it might be unmarked. The man might have died from being stabbed, or an arrow, or fatigue.
There's simply too many variables to look at, and that's not even counting the fact that Mongolian tradition is against violating ancient tombs.
Some sources and extra reading:
On a related note: is there any credible source to the tale of his burial, and the mass executions that followed it to ensure it was hidden? I also remember Stephen Fry recalling an anecdote on QI about them slaughtering a baby camel in front of its mother and burying it next to Genghis because she would know where to return to.