Short Answer: We won't have to worry about the Galactic Empire any time soon.
Long Answer: First we have to consider some of the properties of a Type III civilization - or, to maybe put it better, what kinds of things it would do. Some are shared properties of Type II civilizations, but on a larger scale. You can probably list some off the top of your head:
- Advanced interstellar travel
- Large-scale engineering projects
- The ability to capture energy from stars, black holes, and pretty much any other source you can think of.
- Lots and lots and lots of signals
- Colonies on most star systems
- The intelligence to not contact us
You can extrapolate #1 from just about every science fiction movie ever made. But I don't mean anything like the starship Enterprise, or the Death Star. To become a Type III civilization, a civilization first has to reach Type II status. What does that entail? Well, among a lot of other things, shunning war. I know that might be hard to believe, but think about this: Could a single nation get off this planet on a large scale all by itself? A good example is the International Space Station. To get the project going, it needed the combined help of the U.S., Russia, Japan, China, and the E.S.A. We'll need even more international cooperation to get a bit further out for longer durations of time - and that requires international diplomacy and cooperation.
So I can conclude that any spacecraft we find from such a civilization would most likely be peaceful - a passenger ship, perhaps, or a freighter. Such a craft would be hard to detect - admittedly, we have trouble finding anything the size of a planet or smaller, once we get outside our solar system! But that brings us to #2: Large-scale engineering.
Picture the Death Star. "That's no moon!" Yep, and it looks - from afar - a lot like one. Type III civilizations would have the capacity to build such structures - for peace, likely (see #1) - and these would be a lot easier to detect. Wait. Didn't I just say that anything the size of a planet or smaller would be hard to detect? Yep. But a Type III civilization wouldn't be limited to making small faux moons. Planet-sized objects, or bigger (we'll get to that in #3). How could we detect such objects? Well, if they're big enough, we could find them the same way we find (if we're lucky) exoplanets - transits of the primary star, wobbles in the star's orbit, etc. There is another way, though - for specialized creations.
Ah, 3#. Energy-collecting devices. A quick Internet search of "Dyson sphere" should give you a good idea of what I mean (and if that's not enough, you can see Dyson's original paper or see this question). Dyson spheres/rings/any-other-configuration-you-can-think-of would affect the star whose light they were collecting, and we could figure out if the object in question was a Dyson sphere (see the "Dyson sphere" question). But a Type III civilization could transcend that (Dyson spheres are often cited as the trademark of a Type II civilization). Imagine collecting the energy of jets emitted from the accretion disk of a black hole - not an easy job, but still possible.
Up next: signals. Humans have been emitting radio waves since Marconi (okay, he wasn't the only one) turned on his transmitter. Over 100 years later, we're still going at a pretty good rate. A civilization would need millennia to reach Type III status, and should have emitted a bunch of signals along the way. Also, such a civilization would have spread out, and should be emitting them today. Now, the galaxy's a pretty big place. It is tens and tens of thousands of light years in diameter (although there is a debate about where the galaxy ends), so our radio waves won't have reached much of the galaxy. But a Type III civilization should, theoretically, be almost everywhere, and we should have gotten messages (even inadvertent ones) long ago. Unless you think the "Wow" signal was actually a message, there's not a lot of hope. You hit the nail on the head with your SETI remark.
For the next (#5) property, we turn to . . . well, just about any sci-fi show ever made (again). What would Star Wars or Star Trek be without colonies all over the galaxy? Technically, the definition of a Type III civilization is one that merely uses all the galaxy's energy; it doesn't have to be all over. But it's not much of a conceptual leap to realize that such a civilization would actually have to be pretty much everywhere in the galaxy in order to collect all that energy. So we should, in theory, have a decent chance of stumbling upon a colony if we headed to any star. Is that going to happen any time soon? Nope. So there goes that method.
The last property is something of a joke, but also something of a parting message. As Calvin (from the comic strip Calvin and Hobbes) once said,
Sometimes I think the surest sign that intelligent life exists elsewhere in the universe is that none of it has tried to contact us.
But think about it. We're a pretty violent race. Just turn on the evening news. Remember what I said before: advanced civilizations may not want to contact us. So even if we don't receive any messages, or find any Dyson spheres, of fail to find evidence of any kind, doesn't mean that we're alone on the Milky Way. Unlikely? Yep. But plausible? Yep.
So, to summarize: There are a bunch of different properties that would scream "Advanced civilization" at you. We could detect them by looking at their effects on stars, by searching the sky for radio waves, or by simply hopping over to a few nearby star systems. But being as we really haven't found anything yet: We can conclude, with reasonable certainty, that there is/are no Type III civilizations in our galaxy.
I hope this helps.