Law professor Adam Winkler has a TNR piece today arguing...well, he doesn't quite argue that the filibuster is unconstitutional, but goes through the case against it and sort of suggests that it is unconstitutional but that there's nothing that can be done about it outside of the Senate.
I agree with that last point: the courts aren't going to step in. But, once again: like it or not, there's really just no way to get around "Each house may determine the rule of its proceedings." If the Senate wants to run things so that bills and nominations need 60 votes, they can do it.
I'll also point out, as I tend to do when this discussion arises, that those who say that filibusters are unconstitutional because majorities must always rule are out to protect only certain majorities, not all majorities. That is, the entire committee system as well as the party system within both chambers of Congress mean that only a very small percentage of all bills get votes at all; a well-placed minority can spike any bill. Indeed, filibusters have been used precisely to force a Senate floor vote on some measure that the majority party doesn't want but that a majority of all Senators supports.
And while the current (post-1993, and especially post-2009) filibuster regime is dysfunctional, it's hardly the first time that a Congressional procedure has denied parties the ability to act. The House reforms of 1959-1975 were a reaction to the dictatorship of committees run by chairs who got their jobs strictly by seniority. That included the House Rules Committee, which could refuse to allow any bill the chair didn't like to advance to the floor, and in the short run there was nothing at all anyone could do about it.
So, yeah, reform the filibuster, but really, it's Constitutional. And no legislative body runs by actual majority rule; the question is the extent to which the majority party is the particular majority that gets to run things.