3 Days in Bucharest, Romania

IMHO 3 days is more than enough for Bucuresti. You can go by train to mountain resorts Sinaia, Predeal and there is one between those two which name I already forgot. I was there still under Ceausescu. Nearby Brasov is a pretty town worth seeing.

Ilja's opinion isn't just humble, its generally correct. I sort of like Bucharest in a quirky kinda way, but most aren't like me. So.... you said you had 3 FULL days in Bucharest which implies you are arriving in Bucharest the day before your first "Full Day" and you are departing the day after your third "Full Day". Then this suggestion work.

First, you cant do this with public transport. Well you can, but it just wastes too much time. Call or write Daniel at http://covinnus.com/ and procure his services:

The night of arrival go out and see the People's Palace or now they call it the Place of Parliament. If you don't know the history of Cold War Romania the building is "Impressive", if you do know the history then the sight is creepy at best: http://westsiderepublicans.com/wp-content/uploads/2010/12/33_palace_Romania.png
The next morning head for Peleș Castle http://visit.peles.ro/ and then on to Brazov http://romaniatourism.com/brasov.html for the night.

Day two, onward to Sighișoara http://romaniatourism.com/sighisoara.html and then Sibiu http://romaniatourism.com/sibiu.html for the night.
Day three return to Bucharest with maybe a stop at a village to meet some gypsies along the way..

Day four, depart Bucharest.
http://goo.gl/maps/9HECy

Or Daniel has a canned tour http://covinnus.com/tours/private-tours/transylvania-escape-tour/ that you could have him stretch to three days so you aren't rushing so much.

I expect that Lonely Planet will be helpful, whether you need the whole book or just buy selected chapters.