That depends on the code.

Providing the author did the job properly, all you need to do is recompile it on the target machine.

unlike MS-Windows, not all *nix machines were created equally. A 32-bit program compiled on one *nix computer may or may not run on another 32-bit *nix computer, unless the hardware is identical. There are many different flavors of linux/unix computers and they run many different chip sets (or whatever they are called) (80x88, risk, etc). Programs have to be recompiled for each of these different machine types.

Yes, generally you must recompile code for different archs/non-standard kernels in Linux.

[offtopic]Mac OS X Leopard has a special backwards-compatibility regarding 64 and 32 bit apps[/offtopic]

In some distributions you can, if you're on the AMD64, i.e. x86-64 architecture, since it's a superset of x86 -- there are issues that come up, like the fact that you need thirty-two-bit libraries /and/ sixty-four-bit libraries installed on the system.