Depending on how savy you are, you can get your Verizon S3 working on GSM networks outside of the US. It is unlocked for international GSM use (but still locked for US GSM carriers) and has a Micro SIM slot and support for the following bands:

- SM/GPRS/EDGE 850, 900, 1800, 1900 MHz
- 3G UMTS/HSPA+ (21 Mbps down, 5.76 Mbps up) 850, 1900, 2100 MHz
- 4G LTE 700, 1700 (AWS) MHz.

The issue is that Verizon and Samsung have disabled the ability to change/program the APN for the network you're on. This means it's likely that by inserting a local SIM into the phone you'll only get Voice and SMS but no Data. The workaround it to root your phone, install an app like HiAPN, use Titanium Backup to make that app a System App, and then change/program the APN for the local carrier. I'm not sure if other Verizon Android phones with "Global" support require you to jump through the same hoops.

An easier alternative might be to pick up a Motorola Moto G (Global Version) for $180 (Amazon) for use on your trip. After the trip you can keep it for future travels or sell it on for very little loss (and maybe even a profit based on current eBay prices). Now you can just slide in the local sim for the country you're in (or use something like OneSIM for the whole trip) and you're off and running.

If you do buy a Global Moto G and decide to sell it after your trip, PM me