In my 10-year-plus AD&D campaign we rarely use tiles and never use minis at all. But we did find that simple maps are very useful. Not detailed records of the adventure, or complex cartographies of everything in the world, but just really simple drawings of the location a combat scene was taking place in, for instance. Throw-away sketches, basically.
You will find that it is not necessary to draw really detailed maps. But if you still want players to be able to take advantage of situational modifiers, fictional positioning, line of sight, etc (which is a good thing), it still helps to have a general idea of where things are located and how big they are.
For this my group has started using a whiteboard and magnetic markers. Draw the room, sketch some objects, ask the players to put a mark where they are, and wipe the whole thing clean when it's over. Save a tree!
As far as time is concerned, it's not as important as some might think. Movies frequently compress time when boring stuff is happening and stretch it when exciting stuff is happening (especially to the hero). Your players won't fret if you do the same thing.