I see a lot interesting and pragmatic points mentioned on this thread, but nearly all of them seem to opine based on the short-term consequences of the following: taxation issues, access to jobs, travel restrictions, and education.
Taxes (stupid reason, unless you're extremely wealthy):
Speaking as a former Ex-pat living in the UK, I can tell you that taxes are absolutely annoying as hell, but the honest truth is that the penalties are not bad at all even if you're late. It's a minor inconvenience, and the reality is that taxes are usually even owed unless you break 6-figure salary. You can inquire about Foreign Tax Credit or Foreign Earned Income Exclusion. Taxes is a poor reason to deny your children citizenship opportunity, especially where there is a good chance that such issues of taxation will be harmonized in your childrens' lifetime. Stupid tax policies don't last forever. Unless you're kids make A LOT of money, taxes won't really be a huge issue. If you're living in a low taxation zone, then there might be an issue, be European taxes tend to wash out any US tax claims. Additionally, while some dual-taxation specialists are very expensive, I found it far better to hire a reputable accountant in the US who can properly fill out FTC or FEIE forms, Turbo Tax Pro makes this very easy. I found a great guy who did 6 years of taxes for $650. For me thats a bargain with all my tax complexities.
Access to Jobs (very good reason in favor of citizenship):
The best reason I can think of is to provide citizenship to your children is so they have access to the US job market in the future. People give their kidney's for a green card for a reason. The ability to say you're a citizen helps in so many industries. It also allows you to demand higher pay in some instances. In my case, the equivalent British hire would have been hired for one-third less at my firm.
I found that US citizenship helps with travel freedoms. I have no basis to compare US with Swedish/EU/Belgian travel freedoms but I would wager they are equivalent or superior in some instances.
If Belgian/Swedish/EU healthcare access can be maintained while accepting US citizenship, then its really the best of both worlds. I miss the UK healthcare system dearly. One of the few known benefits of the NHS UK is that even if you live in another country, if you get critically ill, they will come out and fly you back to the UK for medical treatment. Chartered plane, nurse, the works. It really is amazing in the grand scheme of things. Not sure how NHS compares to other EU countries, but I'm sure its comparable and far superior to the US in terms of care and management.
Education opportunity and the ability to qualify for financial aid is the top reason, in my mind, why you should provide your children citizenship. As US citizens, your children will qualify for numerous grants, awards, and scholarships, as well as need-based financial aid. Whether you choose to use this for your children's undergraduate needs is your choice, but your children will find it much easier to pursue graduate studies in the US as a citizen. Foreign students simply cannot afford US universities unless they have strong family support back home. Additionally if your fear is that you don't want your students saddled with debt, don't worry, because the government can only take a maximum of 10% of Adjusted Gross Earnings in a given year. For people with children and average paying jobs, usually this means less than $500 to zero dollars. The great thing about federal loans is that you only really pay if you can afford to pay back. After 20 years of ontime payments (even if payments are zero), your debt is forgiven, and you just pay the tax on the remainder.
So all this said, if dual-citizenship is the option, go for it, besides the taxation issue, I see no serious downside. If you have to give up citizenship somewhere, then the issue deserves more thought. Regardless of all this, speaking to a lawyer is worth it.