Though the consistent success of BioWare's role playing titles has set the standard for game morality models, games before and since have experimented with other methods that, like Heavy Rain, arguably avoid the dominant moral strategy problem.
One example is Tactics Ogre: Let Us Cling Together, the predecessor to the better-known PlayStation classic Final Fantasy Tactics. Like other titles in the Ogre Battle series, Tactics Ogre has a broadly-branching narrative with potential for dramatic variation in events depending on the choices the player makes at certain key junctures.
The first such juncture arrives when the player's allies decide to massacre the citizens of a small town and frame their enemies, intending to solidify support for their rebellion and make it possible to ultimately overthrow the current tyrants.
The player has the option to either go along with this plan or oppose it. This choice dramatically changes the course of the game's middle section and the choices the player will face later. It does not sacrifice any mechanical or gameplay benefits or advantages that the player has already earned.
There are different recruitable characters and items in each path, so players who prefer to look ahead via an online resource (or who have played the game at least once before) might choose based upon these factors, but they are non-obvious. Also non-obvious is the amount of separate content in each path.
Choosing not to participate in the massacre, for instance, technically cuts the player off from content, but also opens an equal amount of content.
The fact that the gameplay is largely linear, moving from mission to mission without much room for deviation, hides this fact better than a quest log, where the lack of a quest (or the presence of a quest the player might choose not to complete) confirms the presence of content yet to be experienced. The player, therefore, does not feel obligated to perform missions that are counter to their moral direction for the sake of the content.
More subtle (and more recent) is The Witcher, another title in which the player must make several choices that have far-reaching effects. These choices are important, but won't change the course of the entire game; however, the specifics of key events, and the characters who remain present throughout the game, can change dramatically over the course of the main, relatively static, plot.
Though the game is an action RPG with many similarities to some of BioWare's titles (not insignificantly, The Witcher uses a modified version of the engine created for BioWare's earlier Neverwinter Nights), it lacks any quantified morality elements.
The choices are not so black and white: an early situation involves an angry mob who wishes to burn a woman they suspect is a witch for cursing their village. The player can choose to side with the mob or the witch, and may well have to make this decision without knowing the facts. This choice will affect the immediate situation, of course, but will also have consequences, good and bad alike, much later in the game. These consequences are not based on an overall morality score, but on the specific choice that the player makes.
Part of the reason this functions reasonably well is that the player character, Geralt of Rivea, is given more definition than in many role playing games. He has an existing personality and morality, and any options that the player has in the game's many conversations are within a certain range of Geralt's grim pragmatism.
But whether by accident or design, this does not get in the way of the player making these choices for Geralt; there are not situations in which Geralt would obviously take a certain action, and often he appears to hate being put in a situation where he has to make such major decisions - but he's also able to rise to the occasion. (We can see a similar trend arising in Mass Effect's Commander Shepard and Dragon Age II's Hawke.)
But all of these solutions are, at a certain point, intractable. Branching narrative multiplies the need for content rapidly. High-end games like Heavy Rain, as a result, are generally quite short for one play. Titles like Tactics Ogre, which uses 2D sprites and limited character animations, are able to pack in more content, but still can only branch so much before the game would be nearly impossible to complete in a reasonable amount of time. The Witcher has the same problem.
Furthermore, as we can see, all of these games make substantial use of characterization for the main character, rather than leaving that character open to be defined by the player. The player can certainly shape the characters somewhat, but they are not able to make a character from scratch, however they want, as is popular -- and increasingly expected -- in role playing games. This will not drive players away in droves, by any means, but ideally significant moral decisions could be implemented with the necessity for such an avatar.