Furthermore, we're not measuring "things", we're measuring temperature. If a "thing" was blocking the CMB, it would do so by being a hotter thing. Thus, a "thing" can't create a cold spot just by obscuring the CMB... it can only create a hot spot, relative to the CMB.
Another place you can see this is in "dark matter" replacement theories. As unsatisfying as an invisible, undetectable particle is to answer the questions about the universe, it isn't enough to just explain one or two subquestions; you need to provide a better explanation of everything we see, including the places where it really, really looks like dark matter is the only feasible explanation (where galaxies collide and the visible matter appears to have gotten detacted from the dark matter halos, as seen by gravitational distortions, which is really hard to explain with any of the alternate theories).
It's "easy" to beat the consensus on one detail; it's proved pretty hard to beat the consensus across the board, even as the consensus is known by everybody to be unsatisfactory in some ways.
For "near by "objects on can use parallax: look at the object on a given date then again 6 months later and see how much it shifts against the back ground stars. Simply triangulate using the known distance from earth to sun as half the side of a triangle.
For "moderate" distances one can rely on certain variable stars that always have a known brightness. Then when you locate such a star, note how dim it is and calculate how far away it would have to be to have that brightness level.
For "far" distances one can measure the red shift. That is to say, on the large scale everything in the universe appears to be moving away from us. Further away object move faster than nearby objects. Thus if you can measure how fast something is receding from us you can tell how far it is. The technique for measuring speed like this is to measure the Doppler red shift.
I suppose that if this "axis of evil" were due to a pattern in nearby dust clouds we might be able to see some hints of paralax, but paralax would probably be difficult to measure for something as diffuse and broad as the pattern visible in this data.
It's my impression, for the record, that cosmologists are pretty confident that these variations aren't due to intervening dust. My sense is that the spectrum of the CMB in any given direction is very clearly a blackbody spectrum (the most precise blackbody spectrum in nature, in fact), so any absorption due to foreground matter should be pretty recognizable (and possibly even something they could correct for).