There are a few ways this could impact your credit score, but whether they will have a significant effect or not depends on your situation. Different banks also calculate credit scores differently, and the exact formulas are kept secret, so there is no way to know what the exact impact will be, no matter what. Still, here are some possible impacts.
The first is the inquiries to your credit report. If there is one inquiry for each card then this could be a bit painful. If it's possible for them to run your credit just once, then the impact will be negligible. These fall off after a couple of years, and generally have less impact the longer it's been. If you don't plan on applying for a major loan within the next year or two then you probably don't need to worry about this.
While the accounts are open, they will affect your total accounts number (obviously). Whether this is good or bad depends on how many accounts you already have. Again, if you aren't planning to obtain a major loan before you close them, then you don't need to worry about this. I believe the impact is greater if you have a balance on them, so keeping them paid off may mitigate this factor if you already have a lot of accounts. Keeping them at a zero balance will also improve your ratio of balance to available credit, so this could be a benefit overall.
The last possible impact is the new accounts bringing down your average account age. Even if you have several old accounts, adding multiple new ones is really going to hurt this. Whether or not this will affect you long term, I'm not sure. Supposedly credit cards are supposed to stay on your report for a while after closing them, but the only account I've ever closed was immediately removed from my credit report. I doubt there is a way to predict if this will happen or not.
With all that said, I currently have a bunch of inquiries, too many open accounts, and a terrible average account age, but my credit is still in the excellent range. The most important factor is that you have a solid history of borrowing money and paying it back as agreed. So long as you do that, you don't have to worry about small impacts to your credit score from short term accounts. Unless you plan to obtain a mortgage in the next year or your credit already needs work, I wouldn't be concerned about it.