Fate has its own grid map analog suggested though. It has stuff like dfrpg's city/themes & threats sheets∆ and talks frequently about tracking stuff like scene aspects aspects with index cards. Fred Hicks even has a blog post somewhere explaining how you can pretty easy use a grid map with fate.Use your imagination and make stuff up. But I play a lot of FATE where the players are encouraged to add things to the combat without asking the GM or needing to look at the map to see what’s available so that ports over to D&D. In fact, I find I have the opposite problem with minis. Players make the assumption that, if it’s not on a map, it doesn’t exist. For me, It creates more restraints and am less likely to make it up. If you’re using your imagination, there’s nothing limiting your expectations.
∆most fate flavors have some flavor of themes threats and high level themes/threats aspects that often go with a sheet of some form but dfrpg's makes a good example since it doesn't require much context to see it's use