Can I use iPad 1 or iPad 2 accessories (such as cases / covers) on the new iPad (3)?
It depends. The iPad (3rd Generation) is slightly thicker than the iPad 2, but there are reports that many cases designed for the 2 will work on the new model. Both of these iPads can use the smart cover technology too. Check with the case manufacturer or online forums to be check the compatibility of specific models.

Update: It seems not all Apple Smart Covers are created equal. Early smart covers may not work with iPad (3gen) because Apple changed the polarity of the sleep sensor magnets at some point during the production run. 

The original iPad was the thickest of the lot and had no smart cover capability. Most cases (if not all) were not interchangeable between the newer models.

Accessories should work unless they're truly dimension critical or require a technology not present in a previous model.

My inCase Origami case and the Apple Bluetooth Keyboard it protects continues to work with all iPad models, as do my car chargers, video adaptors and the Apple Camera Connection Kit. The OtterBox Defender case on my original iPad won't doesn't work on newer models.