In the 1948 film adaptation of The Treasure of the Sierra Madre, the Mexican bandit leader named "Gold Hat" says, "Badges? We ain't got no badges. We don't need no badges! I don't have to show you any stinkin' badges!" That feeling sums up my attitude about grids in tabletop roleplaying games.
The most effective story is not the one you tell to someone, it is the one the person tells himself. Another name for this is the theater of the mind. Dungeons and Dragons offers the opportunity for immersive roleplaying. Players become attached to the character they created. The attachment is not to the figurine they use to represent the character. The attachment is not to the sheet of paper with the statistics representing the character. The attachment is to the persona and history of the character.
I create maps to organize the adventure. I enjoy creating maps with Campaign Cartographer. In the past, I created both square grids and hex grids for the maps. I found the grids cumbersome and detracting.
Denver is an example of a real-life city with a grid problem. The oldest part of Denver follows a grid oriented diagonal to the four cardinal direction. The rest of Denver follows a grid oriented to the four cardinal directions. This means there are a large number of streets with forty-five degree turns. Denver does not line up on a simple grid. The Flatiron building in New York emphasizes the non-rectangular orientation of the streets.
I can draw a section of a map, then pick up the section and rotate the section an arbitrary amount and join that up with another section. Creating a scenario similar to Denver. The map should not be the focus of the adventure. The map helps me organize the game scenario. My goal is to create an immersive roleplaying adventure for the players. My focus is on the theater of the mind.
Think back on your best roleplaying adventure. I doubt the details of the layout of the grid, the miniatures, or the map are the most memorable. What is memorable is the actions of the characters, both player and non-player characters. You may remember, "I rolled four twenties in a row" or "I rolled four ones in a row" but few will remember, "I rolled a seven, a fifteen, a twelve, and a nineteen". The reason you remember a one or a twenty is they have significance to advancing the game.
If a player says, "my character slides up against the wall trying to hide in the shadows." Does the grid matter? What matters is the actions of the character, the monsters, and the environment.
"My thief slides up against the wall trying to hide in the shadows. Using the heads of the gargoyles carved into the wall my thief climbs up to the second level. While the rest of the party distracts the ogre, I position my thief so I can jump on the ogre's back. Then I jump. With one dagger, I cut off the magical amulet around the ogre's neck. I plunge the other dagger into the ogre's neck."
"As you climb up the wall, the third gargoyle's head opens its eyes, and bites your foot...."