The Clone Wars Era in the Star Wars Expanded Universe has many problems with continuity. Most stem from the fact that the time period has the highest media density of any period in the Star Wars timeline, with multiple cartoon series, video games, novels, and comics packed into a three-year period. As more stories, situations, and characters are introduced, more contradictory elements appear.
The most jarring of these apparent contradictions is the role of Ahsoka Tano, Anakin's young Togruta Padawan. Based on earlier Star Wars media, it seemed like the series began in 19.5 BBY and ended before 19 BBY, when Episode III: Revenge of the Sith is set. And yet there are no indications in that film or later media of Ahsoka's fate, or even of the fact that Anakin had an apprentice at all. Later adjustments to the timeline put the beginning of the series around 21 BBY, but this only raises more questions; surely Anakin's apprentice of two years would be worth mentioning, even if an apprentice of a few months could be easily forgotten.
It will take a few more years for The Clone Wars to end and the timeline to sort itself out, with new books and other media addressing the continuity problems. In the meantime, fans can only speculate about Ahsoka's fate, and have come up with a number of theories. Some seem particularly likely, while others are wildly implausible.
Possibility #1: Ahsoka Dies
Evidence For: The easiest way to get Ahsoka out of the picture is to kill her. She would have to die a suitably herioc death, of course, perhaps sacrificing herself to save Anakin and the other Jedi. Alternatively, perhaps one of Anakin's reckless plans contributes to her untimely death. The subsequent self-blame might explain why Anakin never mentions her later.
Evidence Against: With the beginning of The Clone Wars set in 19.5 BBY and the series' projected run of 100 episodes, Ahsoka's death (if it does take place) must be very close in time to Revenge of the Sith. One would expect, then, for it to be at the forefront of the characters' minds. At the very least, any action on Anakin's part that contributed to her death would be carefully considered by the Jedi Council before they agreed to admit Anakin.
Out-of-universe, one must also consider the show's intended audience. Ahsoka was created to appeal to a younger demographic. Killing her would be too dark for the series and would potentially alienate its viewers.
Possibility #2: Ahsoka is Expelled from the Jedi Order
Evidence For: Ahsoka's reckless behavior and inability to follow orders might lead to a tragedy, forcing the Jedi Order to expell her. She does take after Anakin, who is far more reckless -- but she has neither the sheer power nor the prophecy that Anakin has which force the Jedi Order to keep him despite anyone's misgivings. If she continues disobeying orders and mounting battle plans that kill half her men, perhaps the Jedi Order will judge her too dangerous.
Evidence Against: If Anakin's only apprentice is expelled from the Jedi Order, it will reflect badly on him whether or not it is his fault. It doesn't make sense for no one to point it out when the Council refuses to make Anakin a Master.
Regardless, the Order is so desperate for Jedi that they put young Padawans like Ahsoka out in the field -- and that, in Yoda: Dark Rendezvous, they accept Force-sensitives as Padawans who would normally be sent to the Agricultural Corps, the destination of choice for failed Jedi. If Ahsoka is too reckless, that is an argument for training her better, not for expelling her.
Possibility #3: Ahsoka Leaves the Jedi Order
Evidence For: Leaving the Jedi Order would put Ahsoka out of the picture without anything so dramatic as her death and without the problems associated with her expulsion. As long as she leaves the order of her own volition, it will not reflect too badly on Anakin. Since Ahsoka has no ties outside the Jedi Order, such as family, that might make her leave, perhaps the most likely scenario is a romance that forces her to reconsider her commitment to the Jedi way.
Evidence Against: Unlike Anakin, Ahsoka was raised in the Jedi Order, and knows no other way of life. Other Jedi have left the Order -- such as Ferus Olin in the Jedi Quest and Last of the Jedi series -- but that was following a serious tragedy, something that would have a profound effect on Anakin as well. And it's hard to imagine Ahsoka abandoning her loyalty to the Jedi Order for personal reasons, especially when she knows how few Jedi are available.
Possibility #4: Ahsoka is Reassigned to a New Master
Evidence For: Reassigning Ahsoka solves the problems associated with expelling her. Perhaps the Council decides that Anakin is a bad influence on her and finds her a Master who's a bit more responsible. Or perhaps it's something even more mundane: Ahsoka is reassigned after Anakin is presumed dead, just as Anakin was temporarily reassigned to Ki-Adi-Mundi after Obi-Wan was believed dead. When Anakin returns (in this scenario), Ahsoka stays with the new master, who turns out to be a better fit for her training.
Evidence Against: The evidence against the Council reassigning Ahsoka to get her away from Anakin is the same as the evidence against her getting expelled or dying because of Anakin's mistakes: it reflects badly on him, and so should come up when he joins the Council. If she's reassigned for reasons unrelated to his conduct, however, it makes sense that the issue never comes up.