How I built a startup while traveling to 20 countries

It really depends on where you go.

Many places in the world don't have stable or fast Internet. If you are a 1-person startup, you might be able to get away with it. But, if you have partners back in the US, Skype calls might not work very well in a coffee shop with a saturated 1MB connection.

I tried exactly this a few years back while traveling through Asia and I ended up just getting an apartment and working from there (and traveling every couple of weeks when I wanted to get out).

Coffee shop/public Internet also isn't very secure.