Over the past year, I've traveled more than I ever have before, to London, Shanghai, Amsterdam, Paris, Singapore and Rio de Janeiro, not to mention points all across the US. (I've started the loop again--London last week, and Shenzhen next week.) Shortly before I signed up for this job, I switched my mobile phone service to Verizon, which probably has the best service in the US. However, Verizon uses the CDMA transmission standard, not the GSM standard that's used in most places around the world.
Even if CDMA worked around the world, however, I'd still be stuck with high roaming charges. To get around those costs, I buy a prepaid SIM card in each market, and use it in an inexpensive tri-band GSM phone that I purchased from T-Mobile and had unlocked. I don't need a SIM for every country; for example, my Orange UK SIM works across Western Europe, and roaming charges there are reasonable.
For other countries, I did some research on the Internet to find the local operator with the best prices and coverage. For Singapore, I went with M2, and in Rio, I used TIM. The biggest problem with this approach is that I have a phone number for each SIM card, but I know some people who use Skype to forward calls from one central number to their SIMs as needed. You'll have to pay Skype's per-minute rates for the incoming calls plus the mobile phone operator's charges for local calls, but the combination is still much less expensive than international roaming with a US carrier. In my case, I simply distribute my SIM number as needed. Either way, you'll save a lot of money for a little inconvenience (purchasing a SIM.)