A great many people take their claim very seriously indeed. Maria Vladimirovna is recognized by the Patriarch as the "Head of the Former Imperial Family" and both Yeltsin and Putin have recieved and acknowledged her.
There are several reasons that people object to her claim;
1. The fact that her great-Grandmother Grand Duchess Maria Pavlovna the Elder was a Lutheran at the time of her son Kirill's birth. The heir to the throne must, by law, be born of an Orthodox mother. Maria Pavlovna converted to Othodoxy late, just so that her son would be eligible to take the Throne. It has been written that Nicholas and the Dowager Empress were clear that Kirill was not eligible for the throne. After the assassination, the Dowager Empress refused to recognize Kirill's claim for this reason.
2. That Kirill married his first cousin, Victoria Melita, an act prohibited by the Orthodox Church. This is moot, because as Head of the Church, Nicholas II recognized this marriage, so it should be ignored.
3. That Maria's father Grand Duke Vladimir contracted a marriage of unequal birth with Princess Leonida Bagration-Moukhranskaya, a member of the formal Royal family of Georgia. Before the revolution, a marriage contracted with a Bagration would have been considered morganatic because of the absorbtion of Georgia into the Russian Empire. After the revolution, however, the Royal Family of Georgia has the same status as any deposed royal family: France, Russia, etc.
Also, if you believe that GD Vladimir was heir to "all Imperial Rights and Privileges" than if he says that a Bagration is of Royal status, than a Bagration IS of Royal status.
4. That Georgii Mikhailovitch is a Hohenzollern, and not a Romanov. Georgii's father, a Royal Prince of the House of Prussia, converted to Russian Orthodoxy and took the name of Mikhail. He was accepted into the House of Romanov, and signed papers renouncing his and his son's rights to Hohenzollern names, titles, and privileges. You have to ignore that one too.
That's the basic rundown. What it really comes down to is that if you adhere strictly to the rules as they existed before the revolution, there is not one living Romanov descendant who has a faultless claim to the throne. The oldest living Male in the senior branch of the family is Prince Nicholas Romanov, who only has daughters. In terms of the most royal blood, it is without doubt Maria Vladimirovna, who is related to the British Royal family, the House of Mecklenburg-Schwerin, the Royal House of Georgia, and the Russian Imperial House.
So, in summation, its really up for grabs. If you really want someone on the throne of Russia, my money's on Maria-she's the only member of the family who wants it, and she has worked very hard indeed to establish herself as "Curatrix of the Throne." You have to admire her sense of duty, if not her extreme laquered hairdo.
Best,
Nick