I'm playing the game on Normal, so my experience may be different, but in general, I find that playing conservatively really helps. Reaction fire usually will work if you have enough space for it to be effective. Saving enough TUs for a normal, or even aimed shot (depending on the weapon) can help, too. Other than that, it's generally a matter of sight lines and overlapping fields of fire. Reapers are still dangerous, of course, but if you can save enough time to react to them, they're not as bad.
The map might also matter. Carbines/shotguns might be more appropriate to carry on a terror mission in an industrial area. Rifles have middling reaction speed (like everything else), but shotguns/carbines and pistols have good reaction modifiers. Sniper rifles and MGs, I think, have lower reaction speeds, but that's where having enough space and enough reserved TUs comes into play.