Flash MX 2004 now has the ability to make animated graphic symbols with Timeline Effects.

Flash will create new symbols and add them to your library for the effect.

Timeline effects work on symbols, shapes, and text, but it is best to make things symbols first for file efficiency.

1. Select a symbol on the stage (IT MUST BE A GRAPHIC SYMBOL) or a shape or text

2. Choose Insert>Timeline Effects>Effects>

3. Select from Blur, Drop Shadow, Expand, or Explode

4. You can easily adjust timing, blurring, etc. and preview your changes before you accept them.

You can also have Timeline Effects help you with your tweening:

1. Select a symbol on the stage (IT MUST BE A GRAPHIC SYMBOL) or a shape or text

2. Choose Insert>Timeline Effects>Transform/Transition>Transform

3. Choose your options and preview them on the right.

Some people use Photoshop to create blurred images (Filter>Blur>Gaussian Blur) and motion blurs (Filter>Blur>motion blur) and then use those blurs in Flash behind their symbols or as complete artwork.

Many other FX are done with animation practices and principles. See some samples below:

Multi-image motion: Inserting extra arms and legs to represent frantic motion.

Zip Motion: Zip motion involves breaking apart your character and cutting linear parts out of the character to denote speedy movement. Such as the example below:

Blurred Motion: In Flash this can be produced just by stretching the character horizontally or vertically.
example below (the alphaed motion lines were added to the foot to aid the effect):

Shadows: Created with a black brush, converted to symbols, and alphaed 50%.

You can shadow a whole character animation by selecting everything for that character, copy frames, make a new symbol, paste frames your animation in. Drop it on a layer under your character and offset it in the direction of your shadow. Select the shadow symbol and tint it to black in the Properties Panel, also give it an alpha of 50%. You will need to select custom from the Effects area.

If you need to delete part of the shadow for some frames, select the entire shadow animation frames and select Modify>Timeline>Convert to Keyframes. Then you can individually break them apart and erase the shadow areas you don't want.

The following book has a great chapter on FX in Flash:
Hollywood 2D Digital Animation: The New Flash Production Revolution by Sandro Corsaro & Clifford J. Parrott

Download sample FX