That really happens every game for you? I've had it happen a couple of times, but not like "clockwork."
Some pointers that might help:
1) Increase the size of your standing army. If the AI sees someone high on the score graph but low on the power graph, they'll seize the opportunity.
2) If you find yourself being swarmed and are behind on troop power, don't be afraid to switch to nationalism for a bit and draft yourself some troops. It's easy to draft yourself into oblivion, so don't go too gung-ho, but it might be vital to your defense effort.
3) If you're really outnumbered, forget about Combat II -- get the city defender promotions and turtle up. A handful of city defender infantry should be able to hold off a sizeable stack of cavs. The AI will pillage, it's true, but they're not *that* smart about it. Meanwhile, you'll earn no War Weariness (unless these are newly captured cities) while he's drowning in it.
4) Tied to number 3 -- if you have to suffer through that much pillaging, you might as well take advantage of the situation. After he breaks the back of his army on your city defenders, go on the offensive. Your offensive stacks should have a basically unimpeded march to his cities at this point
5) If you've got infantry and artillery, how far away from tanks are you? Do you have railroads? Maybe consider adjusting your research path: both of these can be crucial for defense. Railroads allow you to quickly adjust your city defenders based on the path of the invaders. Tanks not only pack the biggest wallop of the era, but they can attack twice per turn and can be given the barrage promotion.
From the details you've given here, though, I think #1 might be the key. You sound like you might be a bit of a builder, so don't neglect your military. =)