The internet tells me that people in rich countries exploit people in poor countries.  If that’s true–a claim I don’t deny, since humans are routinely awful to each other–then people in rich countries are really bad at it.  

Just like in high school debate, let’s start by defining terms.  The OED’s first two definitions of exploitation, emphasis mine:

The action of exploiting or turning to account; productive working or profitable management (of mines, cattle, etc.)…
The action of turning to account for selfish purposes, using for one’s own profit.
These definitions lack the moral punch that people seem to have in mind when they talk about exploitation, but since I’m making an a fortiori argument these minimalist definitions will work just fine.  
My big claim: to exploit somebody a lot you’ve got to take away a lot of stuff from them.  A little math to emphasize this point:
Production = What the producer keeps + What the producer doesn’t keep

You can see where we’re going here: The countries we call “poor” are countries where the average person doesn’t produce that much.  Formally, gross domestic product per person is low in these places.  But if the average person isn’t producing that much then there isn’t much to take–there isn’t much to exploit.  There are a lot of reasons why you wouldn’t keep the stuff you make (the glories of voluntary exchange for instance) but if you’re keeping almost all the stuff you make then your exploiters are doing a bad job of exploiting you.  
How would exploitation (or a lack thereof) show up in the data?  One place is net exports: If exploiters were running the exploited countries in a “productive” and “profitable” manner we’d expect exploited countries to have massive quantities of net exports to the exploiting countries.  Again there might be other reasons a nation would run massive net exports but the absence of net exports is evidence of an absence of successful exploitation.  
Let’s see which countries have the most net exports per person.  Penn World Tables: 
[Notes: That’s 2005 real GDP per capita on the x-axis, real net exports per person on the y-axis. You’re welcome to check whether the picture looks different with long run averages; Wiki table here.]
The countries that make the “possibly massively exploited” short list are in the top part of the chart.  Brunei, Luxembourg, Kuwait, Norway, Singapore, a few others.  I would venture that there might be other explanations for what’s going on in those particular countries, but more importantly let’s note the obvious point that few poor countries are in the running–Equatorial Guinea and Libya and some judgment calls.  
This is similar to the puzzle of imperialism: A successful colonizer would make the colony productive and then export enormous quantities of stuff back home.  But instead colonies–real direct rule colonies, not places with pro forma allegiances to a distant queen–weren’t that productive so there usually wasn’t much to export.  

Billions of people endure wretched poverty in unproductive countries.  But all that suffering isn’t paying off that well for people in the productive countries.  

Yes, you should start using the term “unproductive” instead of “poor.”  The first term keeps the attention on the supply side–where, in the long run, it surely belongs.  Focusing on the supply side might encourage the creation of solutions; at the least it will encourage better thinking.  
I’ll settle for “less productive”.