Being able to vault over low walls, clamber up things, scale chainlink fences and dive on the floor would be great. Right now, if a grenade goes off you're dead before you're even halfway down.
Oh, and leaning up against walls would be great for taking cover. Edging you and your team along just under the enemy's line of sight would be great, before jumping over and raising hell.
Having vehicles (and troops!) getting stuck in mud would be cool to, so you'd have to take the ground conditions into account when driving around. In a truck, you'd have to stick to the roads and solid, even ground, and in an MBT you could offroad pretty much anywhere, but boggy ground would get you stuck. In something with low ground pressure (like a Scorpion CVR(T), which exerts less ground pressure than a man, fact fans!) you could go anywhere.
Proper animations for people falling off buildings or cliffs or getting knocked around by explosions would be really good too, I've done a few Max Paynes when a nade's gone off near me but not killed me, so I've gone flying headfirst to the side, but still in the upright, so I ended up shooting at people with my pistol before hitting the ground and magically going upright again