How can I get clear, low-noise shots of things in the night sky without noticeable motion blur, without spending too much money? Is it possible to build a DIY tracking mount? Are there other techniques or devices that could help?
There are some DIY options out there, and not all that hard to make if you're willing to spend some time and have attention to detail. The simplest forms are the "barn door" mounts which are basically two pieces of hinged wood with a screw that it is turned on an interval to compensate for the Earth's motion.
Anyways, Catching the Light has a writeup of all the mount options, including additional details for the DIY inclined. The site, in general, is a very good resource for those into astrophotography.
Another technique is stacking multiple short exposures.
There looks to be a good write up of it here.
There's nothing to say you can't combine this technique for longer exposures as well. You could build a simple "barn door" mount, such as John Cavan recommends below, and then stack several 5-10 minute exposures (since the simple barn mount as has a maximum tracking time of around 10 minutes). Not sure if it'd be worth it, but it wouldn't hurt to try once.
UPDATE: Since posting this, I've begun to use DeepSkyStacker - its a great program designed for this. The UI takes a little getting used to but overall its much easier to use for these kinds of things.
Unless you have the technical prowess, the time, and yes...the money (as it would still be expensive) to build your own tracking mount, you probably won't be able to do long-exposure astrophotography.
You can avoid star trails, and do "short-exposure astrophotography" by using three things: Very short focal lengths, very fast lenses, and very high ISO. You should be able to expose up to around 25, maybe 30 seconds before you get noticeable star-trailing.
The wider the lens, the less star-trailing you'll get, since the motion of the stars across the sky accounts for less sensor space the wider you get. A faster lens, obviously, will help you expose a brighter image in less time. CA and other optical aberrations can really kill detail on lenses used wide open, so higher quality lenses that correct for optical aberration will be a huge plus.
Finally, a camera with good high-ISO low-noise performance will be essential. A full-frame sensor is a big plus here, and the Canon 5D II is a big favorite of a lot of CMOS astrophotographers. ISO 1600 an ISO 3200 are not unheard of when doing short-exposure astrophotography. Just be wary of noise reduction in post...stars are true point light sources, and noise reduction algorithms can eliminate stars along with noise if you are not careful.
Depending on your goals, you may be able to get away with higher ISO than you might otherwise. If you are taking still shots, there is only so much you can do before you start losing useful detail. If you are shooting time-lapse, then you can probably get away with really high ISO without it being a huge problem, as the size and detail necessary for video is a lot lower than it is for stills.
If you really want to get crisp, clear, saturated stills of the night sky, then you really need a tracking mount. The cheaper Altazimuth mounts will get you stable tracking for shorter periods of time, however the more expensive equatorial mounts are really necessary to get stable, lengthy tracking at the earths surface. Any exposures or stacked exposure sequences that span an hour or more will probably need an equatorial tracking mount.
This is one of those rare cases where DIY won't save you enough to be worth the bother even if you're pinching pennies. Whether or not something like the "barn door" arrangement suggested by John Cavan would be effective depends on how long you intend your exposures to be.
The principle is simple enough, but the parts aren't going to bring you in at substantially less than, say, the Orion EQ 1 mount/tripod plus the EQ 1M motor. You might be able to DIY something with the motor alone -- that's the hard part. After that, it's just gearing.
For something that's going to bring up the fainter stars, galaxies or nebulae, you're going to need either a very long single exposure or a number of merely long exposures to add together, and something driven by a sidereal clock is really the only way to get there. Home-rigging a mount shouldn't be that hard (if you have a couple of tripods, you can set the head of one tripod to match the earth's axis and mount the other tripod's head on that, with the tilt locked and the pan driven by the motor), but it may be worth spending the whole couple of hundred bucks to make it easier on you and foolproof to boot.