The bible and prayer are really versatile methods of getting His message across though. So when you say "aside from the usual methods of prayer or the bible," I get the impression that you haven't explored the different ways a person has gotten a bit of communication through prayer or the bible. Prayer is an easy one because often there's the element of was the prayer answered if it was a request. Other times though it's a very emotional thing, like how Questdriven's description of feeling an emotion through worship, it has happened with prayer too. From praise, and feeling an even more abundant sence of Joy, because God is sharing the joy with you, to being comforted because of giving our moments of griefs and guilts to God. Another element of communication is always an answered prayer. Expecially if a person recoginize sit and prays a thank you as well. There are other kinds too from reading prayers in psalms or from a person's church or culture or historical figures. Sometimes all there is is a sence that your prayers are listened to, and that's enough sometimes to be strengthened.
The bible too is a work of so much communication. Reading it more, and trying to understand it more is in essence trying to find the communication that He's already given us. I know of a phenomon of reading the bible more then once and getting something more out of the verses read. In my opinion, there's something really true about the scriptures being spirit filled, because it seems like sometimes it feels like when I read it I'm not alone and some verses are just waiting there to jump out at me, and tell me something more. I think God or one of His angles reads the bible with us almost as if it's being read to us as more is pointed out while we read it. Though not always, and not for everyone is it easy or engaging to read. (Some parts can be harder to move through then others sometimes). Then there is also the lessons and the history in the bible. Sometimes when I do something wrong, I don't realize it. But later on I'm reminded of a verse or two in the bible that confront my actions. If you know parts of the bible, they'll stay with you even after you read it. Other experiences like Kiwidan's where needing a solution come from directly opening the bible and reading at any point in it.
But to answer your question, outside of the bible and prayer, I think at least once or trice God's given me a lesson in a dream. Or if it wasn't from God, it was still a lesson I needed to learn at that time. I also think I feel God's presence in praise, and in song. Sometimes through the songs in church I think there's communication to me, simularily like there is from the bible.