I remember reading somewhere(in-game tip screen?) that having pieces of a set will make the remaining pieces drop easier. Is that true? I had a mystical invoker blaze in my inventory for the longest time and the game seems to drop whatever it wants anyway. Do I have to keep it in the main inventory(the one on the left that I start the game with)?
No, it does not affect the drop chance. You can however reroll an item of a set into an item of the set, so if you have duplicates you can use them to get (a chance at) a different item from the set.
It was just someones suggesting such a change for next patch / hotfix 
As said it has no effect on other set items dropping for you.
This is what mamba mentioned