Re: The processor really shouldn't matter to applications these days.
A problem particular to Linux is that the system-call numbers are different on different architectures (notably between x86 32-bit and 64-bit). I'm sure it's not an unsurmountable issue, but fixing it does require the final-stage compiler to know a little more about the bitcode than a straight compiler would.
This would only affect static binaries and the odd thing that actually make system calls directly. Most stuff links against libc so wouldn't be an issue.