If I'm not backpacking or renting locally, I almost always use airbnb these days.
Airbnb works almost everywhere, has filters, has community reviews, and has some pretty crazy good monthly rates.
examples:
Type in Cusco, select "entire home", filter by price under $350 USD, and you have 18 private housing options in Cusco to choose from.
Once you choose a place, that price is locked in and you can extend by a few days or months at a time.
You won't have to repair anything, pay for utilities, bother with anything except living where you want to live.
I'm going to the Philippines next, so I'll probably make a similar search like this: 
and choose the one that looks good.
If you want to live in a more or less expensive area, just adjust the price filter up or down by $50-100 bucks to find a manageable amount of options to choose from.
In Cusco, $350 works. Philippines, $250, France, $600:
That's for entire houses/condos. If you are okay with a furnished room, prices go down. Without maintenance or repair fees, living in Europe, Asia or anywhere is very affordable while traveling.
Is... this an ad for airbnb?
ha, no, this post is about finding housing abroad.
I've used booking, agoda, all that stuff, and find airbnb the best, so that's the housing app I recommend for travelers.