Some options I have used in various games that I’ve liked:
Have purely-nominal alignment
Characters each have an alignment, but it doesn’t mean anything except for whether detect evil pings for you. Super-simple, pretty consistent, but can lead to fairly arbitrary effects.
As one friend put it,
Alignment doesn’t affect anything but the color of your lightsaber.
When you do this, it does not make much sense to enforce alignment-based requirements and prerequisites. Then again, I’d argue that’s close to the default case anyway...
Instead of eliminating alignment, redefine it. I’m a fan of alignment as allegiance; I’ve played a number of campaigns where it’s worked very well. Good alignment becomes allegiance to Celestia, or the kingdoms of man, or the orcish empire for that matter. Evil alignment becomes whichever side is opposed to the previous side. And the same for Law and Chaos. These sides can have as little or as much association with “good” and “evil” as you like (though it’s probably easier to have such associations).
For a game like Planescape, this is very easy. You are Good because you worship a Good god, who is a Good god because his realm is on a Good plane, even if neither you nor the god are particularly good, personally. You are on the side of the Upper Planes and that’s that.
For settings that don’t have such obvious sides, it’s worked best if “Lawful” were the “winners who wrote history.” As an example, this is an adaptation of a game I played in, using Eberron since it’s an established setting that saves me having to explain some of the details and factions of the custom setting.
The human kingdoms in this Eberron game aren’t “Lawful” because of their ethics, but because the Five Nations, and before them the Kingdom of Galifar, have dominated the culture of Khorvaire and have defined themselves as “the Law.” They also label things like cults to the Dragon Below, the attempted revival of the Dhakaani goblin empire, and the monster “nation” of Darguun as “Chaotic” – not so much because of their ethics but because of their threat to the dominance of humanity, which they have already labeled as “Lawful.” “Good” and “Evil” apply largely to individuals rather than organizations: they were whether or not you worked to the benefit of your side even if it is to your detriment (Good), or worked for yourself even if it hurt your own side (Evil). And the overwhelming majority of people are True Neutral because they live peaceably within Khorvaire’s established culture and society, neither actively reinforcing it nor acting to undo it, and are mostly out for themselves (but will stop before really harming those they consider to be on their side) and will try to help their side (so long as it doesn’t put them out too much).
Ban Alignment-based effects
Spells like detect evil and holy word no longer exist. Simple enough, but problematic if people are dead-set on playing alignment-based classes (Paladin being the biggest candidate). Then again, in most cases the class is either more than powerful enough even without these effects (Clerics can do just fine without blasphemy, which is overpowered to begin with), or so weak that opening these effects up more broadly won’t break anything (Paladins that can Smite Anything still aren’t all that impressive).
The biggest problem with this, honestly, is likely to be monsters who use these effects. A pit fiend just isn’t the same threat without at-will blasphemy. Consider changing these sorts of spell-like abilities to be racially-based: the pit fiend gets an at-will ability à la blasphemy that works on non-devils, etc. Considerable power upgrade, though it may not matter much.