I believe that though it was not intended to be this way by our founding fathers, the judicial branch is the most important because of its power. While our nation was founded on the principle of a balance of power, the judicial branch (the Supreme Court)--unlike the legislative and executive branches--does not have regular elections for its positions or term limits like the Presidency. Justices are nominated and confirmed for life seats on the Supreme Court and leave when they choose to, not when they are voted out of office.
Moreover, the Supreme Court is often the last stop for many disputes and Constitutional questions. A current example of this final say and therefore, ultimate power, is the health care legislation. Some opponents have questioned the constitutionality of the bill, and many Senators and Congressmen's answers have been, "That will have to be decided by the Supreme Court." In other words, a bill or policy which the President and the Congress have worked on for almost a year could be permanently negated or overturned by the Supreme Court.
I suppose you could have only three specific answers to the question, but the rationale behind each could be so divergent. For my bet and only to be different, I would submit that the judicial branch is the most important. Being charged with the sole duty of interpreting the Constitutionality of laws and actions provides it with the penultimate check on the other branches. The Supreme Court's ability to possess discretionary review and consisting of individuals who are not necessarily answerable to a body politic and only to the parameters of the Constitution would make this branch essential to the function of the government. To safeguard the Constitution is what could be argued as the fundamental premise behind our system of government.
This depends on what you mean by "important." In my opinion, the executive branch is the most important branch of government at this level.
I believe that this branch is the most important because it is the branch that includes the president. While the president, of course, has no power to make actual laws, I would argue that he (and eventually she) is the most important political figure in the country.
It is the president who has the "bully pulpit" that allows him to set the agenda for the country. It is to the president that people look in times of crisis. So even thought the president doesn't make the laws, he has much more influence on what people think of the government and on what sorts of things the government does than Congress typically does.
So in that sense, I think that the executive branch is most important.