Should a polyatomic crystal behave similarly to the bulk of each/either of its constituent elements? Generally, metals are usually fairly conductive, but their oxides aren't. I know conductivity is just one attribute, but in general, should you expect a, say, diatomic bulk crystal's properties to be anything like the bulk properties of each element it's composed of? Or do all those properties get thrown out the window and overwhelmed by the chemistry of the new diatomic crystal?
 A: Macroscopic properties of materials heavily depend on their chemical(atomic) structure. Even different structures of the same set of atoms can make huge differences; for instance consider diamond, graphite and buckyball. 



All of these materials have vast different physical properties.
So, when just simple rearrangement can make such differences, one shouldn't really expect to get similar properties by adding new atoms to the structure of some element. The new properties are usually far different(as they should); and I would say it's really unlikely that a compound has similar properties to its sub-elements. One reason to consider is, many of the physical properties of materials depend on the electrons on the last orbitals(this is called chemistry), and during chemical bonds those electrons are affected most.
Another thing to mention is, deriving macroscopic properties of material, based on basic principles of quantum mechanics(molecular dynamics), in most cases is computationally infeasible using the current computational power. Using quantum computers,(if we build a large enough one) will help us with this manner as well.
A: So you're basically comparing a crystal of, say, Silicon and that of something like Silicon Dioxide?
The exact properties will depend on an awful lot of things, and the exact design of the crystal can be varied to produce new properties. 
The geometry of the final crystal will have a large impact on the electronic properties. The new atoms will cause a change in the flow of electrons based on their own properties. You can use new atoms to limit the flow of electrons in certain planes, and semiconducting effects can also be observed. Other properties that will effect the electronic properties include the new bonding types, as less free electrons may be present. 
Magnetically the new atoms may affect the dipoles/domains present, but I didn't pay attention in this part of my Crystalline Solids course. 
Optically you will see different results too. Taking diffraction as an example if you start adding new elements into a crystal the peaks you see will vary based on the atoms, their positions, their respective dipoles and how the react to the wavelength light you put in.
Thermally their melting/boiling points may also change due to the changes in the chemical bonding present. This could work in either direction, and is due to the bonds that are formed between the new molecules. 
I may be wrong, but I believe that Iron and Steel are good examples. Once the iron lattice has the carbon atoms inserted this causes a change in the strength of the materials. Iron is quite flexible whereas steel becomes extremely rigid. 
That's a quite broad response, and I'm sure other people can give more detail in their answers. 
A: It's all about the electronic structure. "Electronic structure" is the term for the available energy states and transitions of electrons in the crystal.
Largely, the nuclei in a crystal are much of a muchness; big positive things that don't move a lot. The electrons, however, can move around to different degrees depending on how many of them there are, the band structure (allowed energies) and so on.
Consider the metals - they act fairly similarly; heat and electricity are conducted, they are malleable, they melt at some temperature, they are shiny. Or the insulators - the are not shiny, they are thermal insulators, they don't conduct electricity, they have a dielectric breakdown at some potential difference.
Finally, the semiconductors! They conduct some electrons, but not all, and not in all circumstances. The electrons have different allowed states, and different allowed transitions. Putting slightly different semiconductors next to each other permits interesting interactions and conditional conduction (PN junction, transistors). They can form wells to trap electrons, channels to conduct them, couple electrons to photons, react to charge and form waves of charge along their surfaces. 
You can probably guess which is the fun group of materials there.
Semiconductors can be made from a single material (Silicon, Germanium etc) or from combinations of metals and insulators on either side (GaAs, InAs, GaInAs, etc etc). The results are a series of materials with subtly different properties, all down to the subtleties of their electronic structures; direct or indirect band gaps, (an)isotropic effective mass, recombination modes. 
You can guess some of the properties of a compound material from its components, but usually only whether it will be an insulator or a conductor or a semiconductor. Beyond that, the electronic structure is a complex beast and there are large fields of research devoted to better modelling of it with increasingly complex techniques. For reference in that area, look into Quantum Chemistry and Density Functional Theory.
