There's no requirement that you have to design cross bracing to only take tension; however, due to the space requirements and the fact that each brace will handle movement in one direction in tension, it's often most convenient to do so. Quite small rods and plates can handle the bracing loads in tension. Once you decide to try to add compressive capacity to these members, you usually end up with a significantly larger member to avoid buckling. As such, it's simply more efficient in many cases to design tension only bracing members.
For example, it might not be uncommon to see a cross brace made of 20 mm rod (or M20 threaded rod even) and the length of each bracing member might be 6 m or even greater. The compression capacity of a 20 mm rod at such a length is essentially nothing at all and yet it still boasts a reasonable tensile capacity. Two opposed (X) braces will form a satisfactory system where a single member would need to be quite a bit larger to offer any compression capability.