Artillery. No doubt about it.
My last game was me (japan) on regent. I was at tech parity with several civs, but the blasted babs were at least five steps ahead of me. My goal was a diplo victory, but I knew at the current rate the babs would hit Fission LONG before I got to it, and they would start the UN before I could. The key to a diplo victory, afterall, is controlling the UN vote...
Anyway, the babs had some tough units (paratroops and marines), and some well defended cities. I took my cavalry and infantry and started a world war (advantage of a strong diplomatic network

) against the babs. I made sure that my main attack stack had 20 artillery in it (along with 40 cavalry and 20 infantry). When the stack got to a city (parked on a mountain or hill outside of the city), I would spend time blasting the snot out of the town, using the infantry to defend the stack if it was attacked and the cavalry to take the offensive. By the time I knocked the city down to size 6 (about three turns later) I had decimated their main army due to the AIs methoeds of counterattacking. Units would approach the stack, get hit with an artillery blast, and then get mopped by the infantry. Losing that one hit point sometimes meant the difference between winning and losing.
(The babs were eliminated by the persians in this war, I sued for peace after 20 turns, keeping four cities and gaining most of their tech lead for myself as spoils. This left the Persians as the only civ ahead of me in tech. Hmmmmm...)
If you are striking out against their units, carry artillery with you. Those marines entrenched on the hill may look threatening, but take them down to one hit point with your artillery and your inferior units will have a better chance to take them out. I gain many leaders using this tactic.

Wear down the enemy, then toss your elites at them.