The two rules, off the top of my head, most affected by this change are Attacks of Opportunity (AoO) and Flanking.
Player: Okay, I move to flank him.
GM: Sorry, there's no flanking because there's no grid.
Player: Can't I just move to flank him anyway? My character can see what's going on.
GM: Oh yeah. But you can't see what's ahead of you.
Player: Fair enough.
GM: Okay, when you move to flank, an enemy jumps out of the shadows ahead of you.
Player: You said I couldn't see ahead of me.
GM: Yeah, that's why the enemy ambushes you.
Player: Oh, I was gonna make a joke about how I couldn't see the enemy ambushing me, but... okay. How much damage do I take?
GM: It wasn't a very funny joke. You die!
That'd work, wouldn't it? Just make it impossible to flank when the group is close together, unless there happens to be lots of enemies.
Also, how would you work AoEs? That's one of the main benefits of a grid.
Roll the number of enemies affected and group enemies - if some enemies aren't part of the same group as other enemies, assume that they are attacking from a different direction than the AoE attack. If players are engaging enemies, then there's a chance they will be effected and have to save.
Ie. 2 goblins and 1 zombie are part of group A, while 2 other goblins are part of group B and attacking from the hallway on the other side. Part of group C is a dragon, in hiding.
Even without maps, you'd still need to keep a lot of notes about whatever lies ahead. If you have a dungeon, you still need to write notes about each room and each object within it and then write up a description of each room for the players.
Take or leave it, that's all I got.
It's possible to run without a map - all the stages of planning are the same (except for making the map). The map just makes it feel more tactical, I guess.
Try to be explicit and consistent with things. As in establish a party marching order and how far apart "arm's length" (5ft space between) or more the heroes will be if they are able to. Try to have some terms that mean specific in game things. Like "arm's length" can mean with a 5ft space in-between. "A long jump away" could mean 10 ft. 'Just at a near target away' could be right at 30 ft. Etc.
Probably easier to say '30ft.'. If the players say something like 'I observe the passage way'. If it's too long, then say 'you have no way of judging'. Players will eventually get the hang of the GM's style, and when the GM says things like 'you can't tell', then they say 'huh, it must be really far away', etc.