I recently installed Debian Wheezy onto a hard drive I later intend to connect to a different computer, though the computers use different brands of hardware. Will this work? Or am I setting myself up for failure?
It will work as long as your initial system is not 64bit and the target 32 bit. If you set things up on a 32 bit system and your target is 64 bit you will have some performance hit.
Make sure you don't install proprietary hardware changes (like NVidia kernel patches) if the systems differ there (but Debian might not have those in the first place).
I have done this with SuSE and Ubuntu, first installing in a VM, then copying things over to a real system.