I completely understand your frustration. Ran into this problem with a my brother's iPad 2 and my friends iPad Air who updated from 7.1.2 to 8.3. Both asked me to speed up there devices like they were on 7.1.2.
There were a ton of options I had to click to speed them up to at least a reasonable amount of performance

.
Here is what I remember I did to there iPads to speed up iOS 8's preformance (hope this helps)...
-turn off perspective zoom on the wallpapers (old way to disable paralax effect)
-turn off recent contacts from appearing on the app switcher
-Settings-> Privacy -> Location services (
turn off the ones you do not need here) -> System Services (
turn off the options that you do not want also here)
-Turn off backround app refresh for the apps you do not need refreshed in the backround (Mysteriously on both the iPad 2 and the iPad air, most of them were toggle back on after the update)
-Turn off auto updates [Settings->Itunes and App store-> Updates] this option updates the apps in the background and, ensure that they are up to the latest verstion
-If all else fails, my sugestion would be to reduce motion. I had to do this for my brother's iPad 2 when it first updated from 7.1.2 to
8.0.2 (one of the most buggiest updates in iOS history) Under Settomgs --> General--> Acessabiltiy--> Reduce motion and toggle it on. It is annoying and frustrating to deal with (in my personal opinion) but, it can be used as a last resort to speed up your device [
I never use it because, it disables all animations]
------------
Overall, I can admit iOS 8.3 is better than some of the previous updates, however, I still am avoiding it on my personal iPad Air and my iPhone 4S because, it will slow down both of my devices (especially my iPhone 4S still running 6.1.3

)
In the end, a update that severely slows down a iPad 3 is unacceptable. Even it limiting the performance of a iPad Air (a 1 year old device!) is unjust:

. Above all, not giving us the right to downgrade is absurd


. All users deserve the right to downgrade to any iOS version that fits there heart. For me, I would not be afraid of updating my iPad air from 7.1.2 to iOS 8 and back to the (for me) legendary 7.1 [This update saved my old iPad 2]
Overall, hope this information helps out and, hope you solve your iPad 3 plight
