Yes, our galaxy rotates around a supermassive black hole - and indeed we know how massive the black hole is. Neutron stars also exist, they are collapsed stars that are not massive enough to form a black hole. They consist only of neutrons (hence the name) and are as compact as the nucleus of an atom, but are macroscopic (big) objects. They are therefore the densest material in the universe and pretty exotic.
So you are right that both of these types of objects are large heavy objects that don't emit light. They are also very localised and so wouldn't show up through their obstruction of light (you'd have to be able to resolve something about a kilometer wide as far away as alpha centauri to see the shadow of a neutron star). So they get around the dust evidence too.
As I understand it there's a few reasons why it can't be black holes or neutron stars. Firstly, and I think least convincingly, is that we can see black holes and neutron stars to some extent. Black holes pull in matter and collect a disk of host gas and dust around them. As this gas falls into the black hole, it's going pretty fast and there are interactions with the rest of the dust that's falling in. This gives off radiation and can be seen. So though we cannot see the black hole itself, we can see the accretion disk of matter that it gathers. Okay, so you could have a black hole without a disk for some reason, or a lot of small black holes without visible disks or so on, but it would be had to justify where all of them come from. Equally, neutron stars spin and there are effects on the objects around them that allow us to get a fair estimate of how many there are.
The reason I don't like that answer is because it relies on complete faith in our pretty indirect methods of detection of these objects. However I am not an astrophysicist so I can't tell you what kind of errors you'd expect on the measurement (estimate?) of the amount of mass that is bound up in black holes and neutron stars.
The other reasons that I find a little more convincing are firstly that we know that the mass of dark matter must be distributed throughout the galaxy, and that this distribution should be relatively smooth. In contrast, a load of black holes would be lumpy. You can measure the movements of stars and use that to constrain the distribution of the mass that they are being affected by. I have been to a couple of talks that have claimed that we can now show that large concentrations of matter are now ruled out by such tests, but again, I'm not an expert on those studies. Also, they tend to rely on large numerical simulations of galaxies to prove their results and people aren't certain about the accuracy of those yet. We probably need more computer power to answer that one conclusively.
However the best reason I know of is that dark matter has to have existed in the early universe. By studying the last light from the big bang we see that there must have been a lot of matter that did not interact strongly with the plasma. Now all the matter we know of would interact strongly, so we need something else. The same matter is needed to amplify small fluctuations in the early universe and turn them into galaxies and stars. Now neutron stars and black holes are both the results of dying stars so our evidence for dark matter precedes the time at which we expect black holes and neutron stars to appear on the scene.
That's the reasons I can think of off the top of my head. Obviously each has ways out and we definitely need to obtain a better estimate of the amount of matter in neutron stars and black holes in the universe, but the weight of evidence seems to say that they aren't the matter we're looking for,