My current game - my magnum opus - has no text planned, in the menus, anywhere, only the title. But a lot of the gameplay relies heavily on the player's ability to interact with the NPCs; they are always talking with each other: the player with the others, the others with each other, the players with himself.
The game is story-centric, so obviously the ability for the characters to communicate ideas is very important. I'm designing the world in such a way so that a character's opinions can be expressed through the nuances in his actions as he does things that have mechanical relevance (such as overcoming a challenge).
Things become a little more complicated when I want a character to share an idea that isn't directly related to his surroundings. For example, if I want to show how a character feels about something another character is currently doing, I can have that character deal with the current situation in such a way so as to express that. Huskies, for example, when they sled will cut each other off and bite and tug in various ways to indicate their displeasure with a running partner's behaviour. I can duplicate that behaviour by having my character change how he supports the others. Think about how much you can tell about your party's feelings towards you in WoW or a coop shooter by their avatars' actions relative to yours. Sometimes you can tell far more from behaviour than voice chat.
(Think about the connection the player has with Curly, in Cave Story, just from having her bounce along through a level with you. Her (possible) death is hard-hitting when it comes, soon after).
Anyway, that doesn't cover the case that you are talking about, which is some way of getting information about a past event to a player through some interface, like a computer log... without text. Environmental story telling is one option. Always use the environment when you can, because it's free. Bioshock does this a lot, sort of.
Another, more effective option is to have one character "remember" an event. The logs in Bioshock are an example; they don't represent a person's memory but a machine's "memory" of a person's log. In Machinarium, the semi-recent indie point-and-click, NPCs will remember events and describe them to you through simple pictorial story-lines that play out in speech bubbles: no text.
If you want an actual machine, the best thing is to create a "hologram" (ala Leia in Star Wars) of an actual character or abstract representation of a character's psyche. The standard written/voice log is the same thing, just in a different format. If your characters can already express emotion in their behaviour then it would be a good idea to use the same methods of expression in the log.
The biggest difficulty is inventing a new way for your characters to communicate. Whatever you do you'll be creating a new way to transmit an idea to the player. I recommend picking something that can be re-used a lot, because the implementation time will far exceed the value of a single, or only several, use(s). For example, all of the methods I have planned to use for my characters' communications are present throughout most of the game, so I can make good use them and train the player in their interpretation.
Simple example: in Harry Potter there is a washbasin that can store memories. If you touch the basin you get "transported" to that memory and live through it as if you were there, but you're invisible. You could do the same thing, allowing the player to move around inside the memory, for example, making it interactive: the player touches the computer screen; the lights fade out, and back on in the same room, or a different room; everything is discolored. People from the past are running around doing whatever they do. Maybe the camera is locked to the perspective of a single actor from whose perspective the memory is saved. The player can be "tied" to that actor, maybe, like they are leashed, allowing them only to travel a short distance away. The character in the memory can express himself through facial expressions etc. If you really want to get fancy, you can start messing with the memory-owning character's perceptions, discoloring certain areas of the environment, highlighting others - things like that - to represent how he perceives events. There's a lot of cool stuff in there.
FF6 does the interactive memory a few times.
The best part is, whatever you build, you can use twice. When you write text you are really leveraging the assumption that your player understands a certain language and is literate. You're also leveraging fonts (which are more complicated than they appear). If you build your own tools for communication, even a few simple ones, you're effectively building a game-centered language, which you can customize to suit whatever needs you invent.
Also, non-text communication opens the door for generative story lines (
): even stuff like handling the play-out of player narrative choice. Though that direction is a whole other thing. Just saying for saying, because it's cool.