Denmark, the Netherlands, and Belgium

Haven't been to Denmark, 2-3 days Brugge, wouldn't give Brussels more than 1 day, and Amsterdam can easily fill several days. Just depends what your interests are.

I was just in Brussels- Bruges and Amsterdam. In Brussels, if you are not making any day trip, 1-2 days is good. Same thing for Bruges you can stay a day more and have a day trip to Knokke and take a tram that goes alond small villages close to the sea and stop in some of them such as Dan Haag and Ostende.
In Amsterdam we spent 4 nights. We did a one day trip to the country side of Amsterdam including The Haag and Delft.

I went to Bruges for a daytrip and one day was enough. Honestly, unless you have the time, don't bother with Brussels. And I'd recommend at least 3 full days for Amsterdam. I've been there 3 times and have always found something new to do.

My own personal answer, based solely on my experience on what I've seen in each country: BELGIUM:
Brussels: skippable, or 1-2 days if you have a real reason to go. The ARAU tours of Art Nouveau Brussels are really worthwhile, the Grand Place is indeed grand, the Magrittes in the museum were interesting, but I'm not sure if that museum is open, and...that's about it. Bruges: 2 days is nice; very pretty and relaxing. Be sure to spend at least 1 night to get the atmosphere away from the daytrippers. Ghent: didn't care for it, but others like it. Saw it as a daytrip. Antwerp: my favorite place in Belgium; highly worth 1-2 days. Tournai - a nice daytrip, if you're inclined (not a "must see," but interesting to see a town with fewer tourists than the four above). NETHERLANDS: Amsterdam: I hated it on my first trip in 1990, but liked it a lot on my recent trip in April 2012. I just spent 4 days there, which was a good amount; you could do it in 3 if pressed. Utrecht: not on many itineraries, but I liked it a lot both times I went. One day is sufficient, but very worthwhile. Keukenhof Gardens: I'm the only one who thought this was overrated, so your mileage will probably vary. Only open mid-March to mid-May every year, and mobbed every day it's open (it certainly was the day I was there). Leiden: I loved this college town, and a day here is well worth it. Delft: I was ready to leave after a half day, and totally disagree with Rick on this one; he likes it, but I think it's skippable. the Hague: some good museums, but not a nice city overall. Rotterdam: fascinating, albeit not what most Americans come to Europe to see. Well worth my daytrip, and I'd consider an overnight in the future. continued..

DENMARK:
Copenhagen: I was there in 1991, so I don't have up to date info, but based on my memories I'd spend at least 3 nights here. Fredriksborg Castle: wonderful daytrip from Copenhagen; the town is just as nice as the castle (and usually, I don't even like castles). Helsingor: just OK; not a must see Louisiana Museum: Good, but not as special to me as to others. So, at a breakneck pace: I'd need 3-4 nights for Belgium (split between Bruges and Antwerp, with daytrips possible), or more if other places there interest you. I'd want 3 days for Amsterdam, and then more for any other cities that interest you (although I don't think you "need" more than a day for the other cities, I spent two nights in Utrecht, and could easily spend an overnight in Leiden or Rotterdam). I'd want 3 days for Copenhagen, plus a day for Fredriksborg Castle and then more days for any other places that interest you (I haven't been to other parts of Denmark at all). Then, you have to allow travel time between these places. I hope all this detail helps, with, yet again, the proviso that this is merely my impressions of these places, and only the parts I've personally been to.

My husband and I love the Netherlands. be sure to go see the Kinderdijk. If you go between March and May go to the Keukenhof Gardens. They are breath taking. We have also visited Haarlem. The church is beautiful. If you go, be sure to visit the TenBoom House where the family hid jews and Dutch resistance during WW2. We have also gone to the Delta Works in the province of Zeeland. . . .pretty amazing.

We spent better than five days in Copenhagen, and surrounding areas, in 2010 as part of a Baltic trip. The one major drawback is it's not an easy trip South into other parts of Europe. We wanted to plan a short trip to Hamburg to visit my wife's cousin but it was really not cheap or convenient to do so. I usually plan trips that cover a specific region rather than hop around. If you want to include both, you will probably want to fly open jaw (maybe into Copenhagen and home from Amsterdam), then add another flight from Denmark to the low countries (maybe Brussels) and work from there.

Wow. Thanks for all the responses. Definitely a lot to think about.

" how much time do I need to do justice to the three countries?" A lifetime for the countries as a whole. But obviously, not so long for individual destinations. I found a weekend to be adequate for Copenhagen, although I could have easily spent much more time there. 2 days is good for a first time visit to Brugge. Great city to visit once, but I've found it increasingly less interesting on each subsequent trip. If you are within 5-10 years of university age, you may find it more enjoyable to stay in nearby Ghent and visit Brugge as a daytrip. For me, however, Brussels has become more interesting as time goes by. It isn't as immediately pretty as Brugge or Ghent, but I find the eclectic mix of museums, churches, public parks and restaurants to be far under-rated. This city takes a bit more effort than just seeing the Grand Place and assuming there's nothing more (a common reaction I see on this website). Depending on how far you wanted to venture out, figure 1-3 days. For personal reasons, I am very fond of Leuven and Hasselt. See these if you have plenty of time, but I wouldn't otherwise make them a priority. Namur, parts of the coast, Ieper, Antwerp, Mechelen and the Ardennes would all be worth visit on an extended trip... actually, if you're going to or from the Netherlands, your route of travel will likely pass through Antwerp anyway, so why not stop for a few hours? I'm running out of space, so won't repeat what others have written about the Netherlands. But if you're there in the summer and you're looking for a beach break, the town of Domburg in Zeeland is one of the most attractive beach resorts I've ever seen in Europe. If you visit in the spring, don't miss the gardens of Keukenhof.

I have been to Belgium twice and each time, i only manage to stick around in Brussels long enough to get my fill of the wonderful art museums and then I am off to Bruges. my first time in Bruges, I stayed 2 nights. We fell in love with Bruges. So it was obvious where we would go on our honeymoon. We spent 5 nights in Bruges. And loved every minute of those 5 nights. We spent 3 nights in Ghent and loved Ghent very much. We couldn't wait to get out of Antwerp; we found it drab, boring, dirty, and far from entertaining; left after 3 hours. I have been to Amsterdam, spent 3 nights there, and loved it very much. I would love to go back.

@ Eli: I guess a rainy day on a sunday will ruin any city that may be liked by the masses. Other than a few museums, nothing was open. We couldn't wait to leave. Plus we are not that big on Rubens. But even with our walk thru the main Antwerp strip, we just were not that impressed. I guess not everything is for everyone.

Denmark is expensive, so just beware that time spent there will be costly. Belgium is not as cozy as NL, so I would spend a day in Brussels and a day in Bruge and then head north to my favorite little country: the Netherlands. Of course, I met my husband there, so.... Maastricht is a great city in the south. Enkhuizen is very nice in the north. Here's my take on Amsterdam: http://www.tidewaterwomen.com/travel-articles/europe-world/item/312-romantic-amsterdam