Andrew H - I will also look at your suggestion of including Vienna and Budapest. I can only travel for a more extended time in the summer, so unfortunately I need to sometimes deal with the crowds. As I generally travel in Italy, I have learned to cope with crowds by visiting less traveled, smaller towns and spending less time in the large, popular places. Are the crowds comparable to the main Italian sites? And would there be a significant difference between travel the last several weeks of June vs. July?
I haven't actually been to Europe in the summer. But, I have heard plenty of anecdotes about Prague in recent years. My friends visited Prague in June two years ago and said it was just mobbed. I'd guess it's not better in July. I was last there in September 2012 and it was crowded enough then. I'd compare it to Venice at its worst. Trying to get across the Charles Bridge anytime after mid-morning into the little maze to the old town square means navigating crowds of people shoulder to shoulder sometimes. But if you are prepared for that, having been to Italy enough, I wouldn't stress over it - just set your expectations, and maybe it won't be as bad as you imagine.
One benefit of including Budapest is that it is known to be much less touristy than Prague. (I was last in Budapest in 2005, but I have heard anecdotes to that effect from recent visitors.) Vienna is probably touristy too, but I don't remember it being as crowded as Prague (I've been there only once).
I'd read up on Vienna and Budapest more and see if they interest you. But if your other option is say to explore the smaller towns in Czech Republic like I did on another trip, then it may come down to choosing between big cities and smaller towns. Some of that is just a personal preference. I think I would enjoy my original Prague-Vienna-Budapest trip from 2005 less now than I did then, because I don't enjoy big cities as much as I used to. And staying more in Czech Republic might give you more time to soak up the "Czech Experience" instead of seeing big cities in three countries over two weeks. There's nothing wrong with either option - really depends on what you want to do.
Yet another option to consider: doing Prague and Krakow. There is direct train service between them now, though it's about a 6:30 hour ride. (also direct flights - but if you are going to fly, you can fly anywhere in Europe really.). I love trains, so the prospect of a 6:30 hour train ride sounds like more fun than tedium - another personal preference thing. Anyway, Krakow is another lovely, charming town, also pretty touristy but maybe less so than Prague. And it is close to Auschwitz, if a visit there interests you at all.