I like many of the ideas brought up; supply lines, sliders, the ability to set none/low/medium/high priority to specific squads, 'reputation' and building specific ways of increasing replenishment rates. I feel that a combination of these ideas would gracefully avoid micromanagement but at the same time add depth to the game and provide the player with an important set of decisions to make.
supply lines; i would see them working along the same lines as caravans (that's what they essentially would be). In order to replenish your plate-armour-wearing-mithril-bastard-sword-wielding heavy infantry squad, you'd need to ensure that you have enough composite materials (plate armour, mithril bastard swords and men) in the first place. Once you have access to the raw materials and your supply lines are safe (probably deserves another thread by itself) we then move to...
sliders; simple and effective. Once the raw materials are in place, the slider will ensure that whatever percentage you set is geared towards repleneshing your army. I personallyt think this mechanic would take out a lot of potentially annoying micromanagement.
Priority; as a previous poster has pointed out, do you really want to waste resources replenishing an outdated squad? Or should you invest more heavily in your up-to-date, elite squadron of bear riders? Obvious really.
Reputation; I like this idea because it can add character to the game. If you have an experienced officer who leads a squad that has taken his name (eg Riko's Roughnecks), it would only add flavour to have that squad attract recruits at a (slightly) higher rate than, say, "Pikemen".
Lastly, buildings (and researchable tech, eg 'logistics') that confer bonuses are always a plus.
However, i do have a question for anybody who reads this; should replenishment of you 'swordsmen' squads depend on how many swords you have in stock? What i mean is, say you are producing swrodsmen at maximum rate (ie you produce 10 swords per turn and are producing 10 swordsmen per turn, leaving you with 0 sword in stock). How do you see this affecting replenishment? Initially i was thinking that as long as you can produce swords, it doesn't matter if you have any in stock.