Shooting a flying bird is tricky. I've not tried many times but it's often hit and miss as the birds are quite unpredictable. Last week I was trying to catch some swallows (I think!) and came back with nothing. They're fast little things!!
The best advice I can give is aim to get the highest shutter speed you can to freeze the action. 1/500th sounds a bit slow to me and you'll likely get blur unless your perfect with your panning. You may need to increase your ISO and use a large aperture F4 or 5.6 to give you a quick shutter speed. On the 350D you should be able to go to ISO400 without inducing too much noise in your shot.
That brings me to my second point. Panning is the key to a decent shot. You need to keep your point of focus in the same place in frame for the duration of the shutter opening. It's best to pick a fixed point on the bird and try to track it as it flies, releasing the shutter at the appropriate time.
On the 350D you should have a focusing mode called AI Servo. Using this the camera will continually adjust focus while you hold the shutter half way down. Pressing it down fully will trigger the shutter. AI Servo is a big help in keeping a moving subject in focus. You can set the focusing to the * button as well which I find helpful on my 20D.
Another point I've just thought of and am now adding is that it's easier to photograph the bird as it comes towards you rather than as it passes as the relative movement is less.
Using these techniques I was able to achieve the following shot...
For this I used a shutter speed on 1/1250s @ F/4.0, ISO 200.
Hope this helps?
Cheers,
Steve