One of the strengths of using combat zones instead of squares is that you can easily customize the tactical complexity from combat to combat without the introduction of different rules. I would personally still use a battle map because (a) I invested in one and minis, (b) it helps visualize the action, and (c) it records placement so I don't have to remember as much. That said, it'd be nice to occasionally just hand wave and jump into the action.
Here is a complex map (from page three of the download) that envisions two-story tavern. There's a handful of zones with different traits, the ability to run or climb upstairs (and of course jump back down), and a bunch of semi-evocative zones that hopefully inspire creativity, particularly in using the combat stunt maneuver. Boiling soup in the kitchen? Spooking a horse in the stable? Lighting the hay on fire? Throwing someone out a window? All good.
I don't mean to imply that all of this isn't possible under current rules, but I probably wouldn't have bothered and would have just made a big room with a few tables. When the tactical driver is squares, the number of squares in the room matter. Too few, and people get upset that they can't use their powers. Way too few and some people can't join the action. Too many and the space feels unrealistic because you've just entered yet another 50 x 50 subterranean cavern... With zones, you just decide if the size of the space is small, normal, or extra large and more on. No need to count squares.
In probably took me about as much time to sketch out the complex battlefield (on paper, the image here took a while) as it would have taken me to set up a one-room tavern fight on a battle map. If we develop a standard notation, it should be quick and clear to communicate.
Despite its simplicity, there is actually a lot going on. Say the party starts in the forest edge, the enemies are in the clearing, and the ruins contains a magic item. The fighter may tactical shift into the clearing to avoid provoking, the rogue may tumble (a hypothetical class power that makes tactical move not provoke) into the ruins to find that magic item, while the ranger and wizard fire from the protection of the forest edge. The wizard has short range spells so he can only target the clearing or the ruins at a range penalty. The ranger's longbow has medium range and so can target either zone. If the clearing had the "large" trait, then the wizard could not target the ruins at all and the ranger could only target the ruins at a range penalty; both could still target the clearing without penalty.
There's lots of reason for people to move to different zones and do so with different types of move actions. There are tactical advantages and disadvantages of each zone. If we develop class powers that let people change zones (say, a druid power that can nullify cover from natural sources), then we might be able to force the ranger and wizard to move zones or provoke. It is probably as tactical as the current game, but abstracted at a level that makes micro-managing less important. You aren't squabbing over a single 5' by 5' square of dirt, you are squabbing over control of the fallen ruins. I think that is an improvement.