Brandon, as the guys have said, yes you can make an all wood tri-lam and you can use TBIII.
I assume you meant to say that you can't afford bamboo.
I use hickory for backing in almost all of my tri-lams. It has to be perfectly straight grained, but it is very reliable. Hard Maple also works, some swear by it. Again, straight grain.
Almost anything can be used for the core, maple, walnut, oak, cherry, etc. basically anything but soft-woods. Ideally something that glues well. Again, I like walnut for that reason. I also use a lot of hickory for cores, it's not the lightest, but it's cheap and when I cut backings, I get some strips that the grain is not good enough for backing.
For the belly, you're looking for the usual woods, something that can handle the compression. Obviously osage, and yew. But also ipe, jatoba, massaranduba, and some other very heavy tropical woods work. Troy breeding built one a few years back that was hickory for the back, core, and belly.
If you're trying to keep it as cheap as possible, and you need to use lumberyard woods, I would say hickory or hard maple could work as a belly, you would just need to build the bow wider than you would wih osage or yew.
Give us more info on what you have access to, what your desired specs for the bow are, and you should be able to get some more specific advice.