I'm not sure that there is a trick really. I have very good spatial imaganition, so if necessary I could keep the map in my head. But that's rarely what I do. Instead I picture the scene in my head, and try to interpret the proposition in light of the scene.
In a typical encounter, the PC's might be trying to rescue the merchant prince's daughter from the bullywog witchdoctor and his body gaurd. I reason that the body gaurd are acting to protect the priest. Hense, the default situation is, "if you want to get to the priest, you have to take some AoO's." Until the PC's do something tactically to open a path, I assume bullywog body gaurds get in the way. Opening a path might mean killing sufficient body gaurds that there are now more PC's than bodyguards, stating that they plan to force a path through line of bodygaurds, or if there is enough room stating that they plan to spend a round in a flanking manuever and another charging the priest. I'll judge the success or failure of such tactics by thier combat success and the number of remaining bullywogs.
Exactly what square anyone is in isn't really the issue. The real issue is who is adjacent to who. So long as you have an idea of who is adjacent to who, you can determine who can be flanked (character has no other ally adjacent is obviously eligible), who is available to intercept (not adjacent to anyone), who is guarded (two adjacent allies), and so forth. You then just sort of decide who can change thier adjacency or whether tactics have changed adjacencies and advice characters whether thier current tactic will draw an AoO.
Sometimes though, I'll play with a battlegrid. In particular, if I've designed the room to have tactical problems in it, I'll want to highlight that aspect of play. If the tactical problem is simple (one or two monsters, or no terrain, or tight quarters), I'll avoid the battlemat.
I did have a 1st edition DM that used a battlemat but put it behind the DM screen so that we couldn't see it. Basically, he was doing as above but using the map to concretely map out his ideas. That might work too.