In particular, they offer three major arguments:
1) The individual mandate falls outside of various Congressional powers typically used to justify the mandate -- such as the power to "tax and spend", the power to regulate interstate commerce, and the "general welfare" clause.(Read the full text of "Why the Health-Care Bills Are Unconstitutional".)
2) The various state-specific deals that were cut with individual Senators to secure the 60 vote run afoul of the general welfare clause.
3) The requirements that state governments establish insurance exchange violates federalism.
I'm not sufficiently knowledgeable about Constitutional law to know whether such challenges would have merit.
And I believe that any long-range effective opposition to ObamaCare will have to be based on fundamental moral and philosophical arguments, as opposed to more derivative Constitution-based arguments.
But I suspect we will be hearing more about such legal arguments in the near future.