I've been running a Swords & Wizardry play-by-post game for the past
year or so and have been using dexterity (DEX)-based initiative. In
the S&W Core and Complete rules, this is the "Blue Book" combat
method, so-called as it is based on the blue covered Holmes Basic
D&D, which described using DEX to order combat actions (while the
1977 Holmes Basic book is not freely available, the BLUEHOLME
Prentice rules are both free and a decent clone of the original).

When combat begins, I'll roll DEX on the spot for the monsters, and
proceed from there. So rolling DEX and ordering the combatants is
done just once - then, during each combat round, actions proceed in
DEX order. In the event of tied dexterities, I consider the attacks
to be simultaneous. The only exception to the DEX order is that
prepared spells are always cast first in the round (this is part of
the S&W rules-as-written). Because the DEX rolls and ordering are
done up-front, after the first round, combat tends to be fast. I've
noticed, however, some problems.

* Luck tends to have a longer lasting impact on combat, when
compared to side-based initiative that is rolled every round. If the
referee rolls high for monster DEX, and you have even average DEX
rolls in the party, you will get stuck going last each round.

* Conversely, if the players got lucky and have PCs with a lot of
high DEX scores, most of the combats will be unbalanced in favor of
the party. This isn't normally bad by itself, but you want the
players to survive combats by using good tactics, not solely through
one lucky roll at character creation.

* OD&D, upon which S&W is based, describes abstract
combat. DEX-based initiative is at odds with that unless you treat
all the monsters of the same type as having the same DEX (this is a
suggested rule in S&W). It feels a little too precise to have an
explicit order for every combatant in each round.

* OD&D also minimized the impact of both high and low ability
scores, but using DEX-based initiative inflates the importance of
DEX.

Compare this to the standard side-based or individual initiative,
where the players and monsters roll every round and the order can
change from round-to-round. This allows luck (good or bad) effects
to happen at any time, and as a player feels more meaningful (or
surprising), in my opinion.

A method I've settled on when I'm running games for my gaming group
is to use side-based, d6 initiative, but allow players with a high
DEX (13 or more) to act first, or to make players with a low DEX (8
or less) act last, but only in the event of tied d6 rolls. This
gives some players a slight advantage (or disadvantage), but doesn't
amplify the affects of a high or low DEX score either way.

If you wanted to keep DEX-based initiative, but fix some of the
issues I listed above, you could use the method described in the
Holmes rules - forego simultaneous attacks for matching DEX scores,
and roll a d6 for scores within 1-2 points of each other, each
round. This results in a lot more rolling (the average 3d6 roll is
9-12, so a good portion of the total initiative rolls will end up
being be d6 opposed rolls with this method, and not all strictly
DEX-based).