How can I safely cut pieces off a small wooden cube?
Jigsaw is a poor choice of tool for this given the size of the finished product and that it is a roughing tool more designed for cutting sheet goods. 

Tools to consider

Bandsaw

If the polyhedron was a little larger and your valued the tolerance of the finished product I would most certainly suggest the bandsaw as you can use a mitre slide and a custom sled with a proper angle to get all of the cuts accurately. However the tiny size hinders the success of that size you cannot clamp that small and that is too close for the fingers. 

Coping saw and/or sandpaper

It might take a little practice but if you are up to using hand tools then I would suggest a coping saw and sandpaper. Use the saw to get the larger sides cut and then just sandpaper to get the triangle faces done. It is small enough to not take too much effort. 



Notes on mass production

As per the comments it looks like you are going to try and mass produce these. I still think sanding would be safer but I suppose you could use a disk sander to speed up the process. Just keep in mind that it is still a power tool that can hurt you so always be mindful of that. Don't sacrifice in an attempt to increase efficiency. 

I would consider making a small ramp of jig that would give you a 60 degree angle. Looking at an edge view from the wiki that can help ensure the best angles when sanding. You can try and use a set of small squeeze clamps to help keep your fingers out of harms way. Those might become less useful as you form your cube since the sides will be made smaller and there will be less to grip. 



Perhaps this product already exists if you are interested

If you do need that many I would look to see if someone already has these machined and sells them. I found this seller on Etsy that makes what appears to be the correct polyhedral. This one as well



Something like a bead supplier could have these and they could likely be finished as well with varnish too. 



Woodworking.SE

If you do nail down a process/procedure based on tools available but still need more help consider coming over to woodworking.se and asking there.