This is a powerful question and one that strikes at the very heart of our legislative branch of government.  In the final analysis, members of Congress may vote in any way they wish.  There is nothing to mandate them from voting a certain way.  The reality is that there are consequences for not voting with their constituency.  For instance, there is much discussion within the recent health care debate that members of Congress are brokering deals and voting with their party affiliation as opposed to voting with their constituencies.  The result of this could be public backlash in the upcoming midterm elections when these Congressional leaders could be voted out of office.  The framers did design the legislative branch so that its members could understand issues in a more clear and focused light than the people they were serving.  Yet, taking this philosophy to a modern extent could result in being voted out of office, as the public is much more privy to issues than it was at the time of the framers and simultaneously more susecptible to media perception and the political "spin" of opposing candidates.  This might compel elected members to vote with their constituencies in order to maximize political survival.

The answer to this is definitely not.

Members of Congress are not obliged to consider their constituents wishes in any way, shape or form.  They are free to vote in any way they please.

The founders of the US did not want a direct democracy.  They wanted a republic in which legislators used their presumably superior wisdom and knowledge to enact laws that were best for the people, even if the people did not agree with those laws.  Therefore, they did not require legislators to consult with their constituents on particular pieces of legislation.

Of course, if legislators go against their constituents' wishes too often, they are liable to lose their next election.  So in that sense they have to think about what their constituents want.  But they are under no legal obligation to determine what the constituents want on a given bill and vote accordingly.

Are members of Congress obliged to vote the way their constituents wish, even if they personally disagree?

No.  Members of Congress are free to vote however they like, regardless of what their constituents want.  It is pretty much inevitable that it would be this way, and it is also, arguably, a good thing that they should do this.

It is inevitable that it should be this way because there is no reliable and cheap way to find out what a representative's constituents think on a given matter.  Polls cannot be taken on every issue that comes before Congress.

It is, perhaps, a good thing that it should be this way because representatives (hopefully) are more informed on the issues than their constituents are.  Sometimes it may make sense for the representative to vote his or her own conscience because he or she simply knows more.

This is not a danger to democracy because voters can, after all, vote their representative out if he or she goes against their wishes too often.

If you wish to research this further, the idea that representatives should vote their consciences is called the "trustee" model or theory of representation.