It's hard to give a realistic answer without the information which mixture has sought (above).
In many countries there are providers offering wireless internet devices with prepaid broadband plans at attractive price. Often the device itself is locked to the provider so buying a different card for each country where the original provider's service is absent may be money wasted. An unlocked device would be a good start.
Alternatively it is possible to buy a sim card which may be able to be used on your mobile phone as a tethered wireless modem - assuming again that it is not locked to the original service supplier.
There are quite a few good deals on offer in Australia which cover most of the populated areas of the nation without the need to change anything. Elsewhere it seems that things get a bit complicated.
This is one area where a genuine multi national service provider could be a good thing. It doesn't seem to happen however.