I've used a middle ground among these answers: I ballpark time when it's not critical, and when it is (in combat, or exploring a dungeon) I track time meticulously.
Ballparking time is easy enough and doesn't require any paraphernalia. It feels somewhat awkward at first, but soon you get more skilled with estimates, communicating them clearly, and generally managing time that it becomes second nature.
Meticulously tracking time does benefit from tools. (I've not used chips/counters, but I expect they work about as well as the solution I settled on.)
For rounds I merely mark their passage with hash marks on scrap paper (usually the same notepaper I track opponent disposition and hitpoints on).
For exploration time-tracking, I've used a few different methods, and none I've tried have worked better than a print-and-assemble paper dial tracker that I used in my last BD&D campaign. (It bundles two variants, one for BD&D and one for AD&D.) It has markings for each ten-minute turn, with additional notches for when a wandering monster check should be made, when rest turns are required, and when torches burn out and oil lamps need refilling.
Using the turn-tracker revolutionised my dungeon-running experience. Rather than having to constantly keep in mind how frequently bookkeeping needed to be done (e.g. making sure to remember "every 6th turn tell them to rest… hm, was that last turn the 4th or 5th since last rest?") and all the other details, the dial simply told me. All I had to do was remember to turn it a notch and announce the passage of time whenever an exploration turn of 10 minutes was completed (which was sometimes ballparked for non-standardised activity), and it reminded me of the bookkeeping tasks I would otherwise often forget and need to fudge later to make up for.
I noticed a corresponding heightened respect for the dungeon's risks in my players too—they were acutely aware of the torches and time passing, and the pressure of the occasional wandering monsters (made in secret of course, without revealing the schedule) gave them a keener appreciation of the risks entailed by spending unnecessary time in the dungeon. Rather than waltzing about the dungeon with relative impunity, they became alert and strategic, being careful at times and rushed at others according to their resources, situation, and goals. Rather than leaving it to me to decide for them what was "unnecessary" time, they could make that choice for themselves, underlining the strategic elements of dungeon-delving and increasing their ownership of their strategic choices. This in turn made me feel more impartial as the DM, allowing them more freedom to chart their own destinies.
It's a pay product, but I found it well worth it. For the price of a small cafe latte it had an disproportionately positive effect on both the game experience and on my own sense of organisation and competence running it.