Thanks to everyone who responded to my last info request. Now, I have tried to purchase an unlocked tri- or quad- band phone for use in Hungary and other European countries and can't find any except on internet. Is that the only source of these phones? Is it better to buy the SIM cards and phones once we arrive in Hungary? Is this also true of other countries in the region? We are a group arriving at different times and countries and meeting in Hungary.

How about Vodafone or Motorola C139?

Thanks!

In most areas of the US unlocked GSM phones are not exactly available on every street corner. You need to shop on the Internet to get the best deals. Today I noticed a Motorola L2 quad-band GSM phone on TigerDirect.com for around $110. Some folks don't like Tiger Direct but I've never had a problem with them. I bought my Moto V551 from them.

Beware that prepaid kits in Hungary are usually with locked phones, so you would have to use that sim-card only and cannot unlock it until you go back home and find someone who can do that for you.
Buying a phone on eBay or internet shops is usually cheaper and easier.