The idea of determinism usually starts from one of two premises: the supremacy of nature or the supremacy of God. Both routes are fairly circuitous, and both involve a leap of faith. But what if we address the question directly?
First, I need to introduce a concept which I'll call "the unprovable". The unprovable is the things which may or may not be true, but cannot be proved even if they were true. The simplest example is this: what evidence would you accept that you don't exist? Not that I, the author don't exist, but that you, the reader, doesn't exist.
The answer, of course, is that there's no evidence which could be presented to you that would lead you to believe that you don't exist; your experience of any evidence is based on the premise that you exist to experience it. There's nothing that you can see, touch, taste, hear, or feel that can disprove your existence; at best they can totally disprove that your senses accurately correspond to reality. (Just to be rigorous, I'm using the standard definition of "reality": that which, when you stop believing in it, doesn't go away.) Thus, while you may or may not exist, to you your non-existence is unprovable.
I don't think that determinism is quite in the category of unprovable, but given how strong the evidence for free will is (how many things in the world seem more true than our ability to choose?), the evidence for determinism should be very strong. In particular, what evidence would actually prove determinism?
Clearly the ability to interfere with the brain and thus with free choice does not prove determinism. If it did, chairs would disprove gravity. The fact that something can be interfered with does not disprove its existence. No, the important feature of determinism is predictability. If something must happen, then you should be able to say in advance that it will happen. This was the position of the materialists of the 1800s and early 1900s. They thought (incorrectly) that the world was basically a big 3-dimensional pool game, and that if you could get precise enough measurements, you would be able to predict the future. They were superceded by the materialists of the mid 1900s who concluded that you can't possible take precise enough measurements (due to the Heisenberg uncertainty principle), and so while the future is determined, you can never predict it.
Some of them presume a restricted form of prediction. That is, they assume that the brain is large enough that quantum indeterminacy can be disregarded and that with precise enough measurements one could predict the brain fully. Of course, we don't have the ability to get precise enough measurements without interfering with the brain yet, so the theory is untested. The test, to constitute strong evidence, would have to be something to the effect to taking the appropriate measurements of the brain without interfering with it, and then predicting, with strong accuracy, every movement that the person will make and every thought that they will think (in some restricted environment, of course) over some reasonable period of time, like an hour. No one's ever done an experiment even remotely like it. So let's be clear: those of us who believe in free will do so upon strong evidence constantly before us. Those of us who believe in free will have yet to produce any direct evidence of their claim. And for many determinists, they can't possibly ever produce direct evidence for it.