S.E. Asia's large choice of destinations and developed internet access systems with a low cost of living and international foreign expat resident population is difficult to beat.
Thanks again to everyone who's commented, and Ricky especially -- I never would have thought of all those considerations for renting and definitely would have made all the mistakes you mentioned!
It sounds like your idea of renting for a month at a time would be the best way to go for me. I do find it difficult to concentrate in cafes and such. What's the best way to find a short-term rental or sublet? Do you look online before you arrive, use an agent, or just seek out local expats (if you don't speak the language) who might know of someone subletting?

I use a range of possibilities for getting apartments to rent for 1 month.
Which one i use, depends upon the country.
Sometimes i will try to book an apartment in advance, but usually i make a shortlist of apartments that look good and suitable, arrive in the destination, stay 2 nights in a hotel whilst i scout around looking at the apartments on my list.
As i say, different countries, different ways of working and booking the apartment.
I NEVER pay a deposit for an apartment in advance of arriving.
What looks good in the photos could be a dump when i arrive... plus, i only have the owners word for it that the wifi internet is good. Personally, i would rather miss out on an apartment than pay a deposit for something that is not what i need.
I very rarely use agents, unless i absolutely have to.
Usually i scan:
craigslist and send enquiries
google search and send enquiries too.
Remember that just because an advert or listing does not mention that that they give a discount for a one month rent, it does not mean it is not available.
Some apartment owners are snobby and won't even think of giving a discount for filling their apartment for one month, others understand how valuable you are as a customer and are willing to negotiate.
If i have confidence in a destination, i might even ask around when i get there too and scan the local papers etc etc.
I stress again that each country is different for renting apartments and how i approach it.
Also remember that some hotels have apartments and suites that you might be comfortable in...some will work out a good deal, others won't.
Don't take adverts and listings as read:
sometimes:
"Cosy" means small and cramped
Every apartment is described as "beautiful" when it clearly isn't.
"rustic" means old
"Homely" means small and cramped, and possibly a flea pit.
I never even consider a listing with no photos. i don't waste my on them. If apartment owners cannot be bothered to try to sell to me, then i don't want the apartment! - lesson learned by experience!
PM me if you decide on a particular country if you want some individual pointers.