What I ended up doing is puréeing my fruit in the food processor, and then pushed it through a fine mesh sieve, a bit at a time. Then I took the remaining pulp and ran it through the food processor a second time, and then back through the sieve. I also added the juice of half a lemon and half a lime. I ended up with about 2½ cups (625mL) of mostly pulp-free, sweet juice, and a container full of puréed mango and pineapple that has the consistency of applesauce (though I'm not sure what I'm going to do with it, and it seems to have lost much of its sweetness).
I tried cheese cloth as well as my sieve, but with the cheese cloth it seemed like I could not do very much at a time (though it was easier to squeeze juice out of the pulp when I could put the whole thing in my hand).
It was a lot of manual work for not a huge yield, however, so I don't know if I'd bother trying it again without a juicer.