This is for anyone having trouble with their turn signals AND hazards on a 2003 Honda Civic DX. This forum helped me piece most of it together, but some info was missing that I hope this will cover for future relay sufferers.
My signals and hazards were doing all these things, in no order:
blinking a few times then stopping
not turning on
blinking steadily, then flickering, then steadily, then stopping.
Internet research told me it was an issue with my turn signal relay (which does control your hazards as well - there was some misinformation about this at AutoZone). Then I had a REALLY REALLY hard time finding that turn signal relay in order to change it.
Here is a picture for you, future reader. May you not spend 4 hours looking for it and trying to replace the wrong relay, as I did.
1. it's in your under-dash fuse box. AutoZone will lie to you and have you puzzling over the one under your hood. Don't listen to them.
2. Pop off that panel just under the steering column. So easy! Enjoy things being easy because they won't be after this.
3. Be smart right now and shake out your drivers floor mat. No. really. Also set up a chiropractor appointment if you do that kind of thing.
4. Get down on your knees on the ground. Slide the drivers seat as far back as it will go. Flip over and lay down on your back with your head in the floorboard next to the gas pedal. Great. That's the only way you can do this.
3. Look at that fuse panel. There's a screw on the lower right hand side. It's screwed on really tight. Unscrew it and take the screw out completely.
4. Pull the entire fuse panel down. At the top right, buried under a lot of cables, is that turn signal relay. It is locked in there by some very firm tabs at the top and bottom. I recommend growing a third arm to do this right, but if that isn't an option, try to just jimmy out one tab at a time. You are going to have to pull damn hard.
photos here: http://flic.kr/p/ag3hHK
and here: http://www.flickr.com/photos/misslacy/6077404821/
5. I didn't test this out, but the AutoZone $10 relay was about twice the size as the one that was a tight fit under the dash (I also read that this was an issue for an Odyssey owner who ended up wasting that $10), so I sucked it up and went to the Honda dealership and paid $20.99 for an OEM.
6. Pop the new one in. Yes, those tabs are going to get in the way again, but not as bad. Screw the fuse panel back into place with that lower right screw. Pop that panel back on.
YOU DID IT! Good job.