IME the easiest way to bring a cutting and have it grow is to take one that's already rooted. These vines tend to creep on the ground and send aerial roots, which hit the ground and root themselves in it. So just find a creeping section, cut it on both sides where it's rooted and dig up the roots. You don't need any leaves, as any root node is also a leaf node and will send new shoots once separated and planted. This also doesn't kill any parts of the original vine as the creeping section that's being taken from has multiple rooting points all along the way.
Vines growing in deep, jungly shade tend to creep more than ones growing in a more sunny aspect. So that's where you need to look for suitable cutting material.
The cuttings take air travel well. Once you've landed, pot them up in a well-drained mix. Don't overwater them. My indoor conditions (~21C and 50% humidity) work fine. I like to put them in a heated propagator at first for more heat and humidity, which may give them an initial boost, but don't think it's necessary. They outgrow it quickly and you can take them out of the propagator without any need for acclimatisation (not with the conditions mentioned above anyway).
Once they start climbing, give them support and they will skyrocket. A young vine can basically be in two stages: initially it just grows like a shrub, but eventually it will send its first searcher shoot (or whatever the correct name is) which goes turning in circles looking for support. Once you have such a searcher shoot, you have a vine and given support it will gain height rapidly.
The transition from stage 1 (shrub) to stage 2 (vine) can take time though; with one of the vines I brought back from my travels in the past, it took three months. During that time it did hardly anything, but once it took off, it hit my bedroom's ceiling within a couple of weeks.
I've had luck with both the tucunaca and caupuri varieties using this technique.
See my post with
Caapi plants that are less than four weeks old.