Well there really is no short answer here. I would do some research by talking to your US mobile provider, looking up rates from European providers, and looking into other options like getting a European SIM card for your phone, or renting a phone.
First I’d contact your mobile provider and see what rates they offer or if they have a partner abroad that they can recommend. If you do use your US phone overseas make sure you know the rate per minute and also be aware of roaming charges. Some companies charge you roaming and international rates for just having your phone ring abroad.
See who the mobile provders are for the country that you’ll be visiting and find out if they have special packages/deals for people traveling abroad. Renting a phone could be a good way to have a phone but not have to worry about local rates. If you have an ISIC card you can go to myisic.com for information on phone rentals and calling card rates. If you want to make calls to the US a calling card might be a good option.
By checking out all your options you should be able to find one that fits yours needs and is inexpensive.

Traveling outside of the US I’ve used a phone with a prepaid SIM card since 2000. They are relatively cheap to buy and will give you a number in country. International calls out are not always cheap, but it doesn’t cost anything to have people call you. However, SMS costs are the same for domestic or international texts, and many people use their mobiles primarily for this.

You can either bring your US phone with you if it is SIM card enabled (check to see it will work on the proper foreign frequencies, and that it is unlocked so you can use any SIM card), or purchase a cheap phone for your trip. I would also look at buying a 2nd hand phone online, perhaps craigslist, again looking for one that will operate for you overseas.

If you are looking to call home while overseas, I’ve generally use Skype since 2006, which is a cheap way to call landlines, or free if the other person also has a Skype account. I think that any web cafe you would go to has a headset and Skype on its computers these days, it was certainly true a year ago.