+1 on OP's comments on Prague. I spent several weeks in Olomouc - a couple hours outside of Prague. I suppose it's not "undiscovered," but it's off the beaten path. It has relatively low prices, great dining, gorgeous churches and cathedrals, and, wonderful friendly people.
I have driven there from Prague, and also taken the train. The Pandolina (high speed train) is my preferred method.
I also had the opportunity to visit Bratislava while in Vienna. Took the "twin cities jet boat." The boat ride was it's own adventure. Don't spend the $$ for first class. The main cabin is spacious and you can walk outside, go to the bar, or grab a snack while enjoying the views.
Slovakia and the Czech Republic separated peacefully after the cold war. The recent history of Nazi and Soviet conquest / occupation will bring tears to your eyes. However, the kindness and hospitality of the locals may just bring tears of joy.
Highly recommend both side trips if in the area!
+2 on Op's comments on Prague. We were in Prague for three days in May. Fabulous but way too crowded. Years ago I did some work in Bratislava and Banska Bystricia. I sort thought Banska Bystricia was sort of neat. I also liked Kiev a lot - but I'm not really sure how to tie Kiev into a longer vacation.
Michael