Hi Eyeshield- my Greek is sadly lacking at the moment, but I did have fairly decent Koine a couple years ago, and I'm happy to tell you what worked for me.
The classes I was taking were focused on translating, but the professor insisted that we read aloud sentences at a time- simple ones, ones we could puzzle out- before translating them. At first I thought he was full of it, but eventually I realized that I would already know what the sense of the sentences were before we started translating them. It was unconscious internalization!
So I suggest reading passages aloud to yourself- ones that are at or a little below your difficulty level- without consciously trying to translate them, and I think you'll start to notice this happening. Also, sometimes when you translate- once you've reached a certain basic comfort level with whatever level of the language you're at- when you sit down to translate a passage, don't go through and do all the parsing and such right away. Instead, read the passage aloud to yourself, and then try to translate the passage off the cuff, in a gestalt. As time goes on you'll notice yourself getting better and better at it. And it will keep your brain from treating the language like a really tricky math problem. (You know- suffix plus accent mark equals pluperfect, or whatever.) That math problem approach has always been the kiss of death for my language learning attempts.
This will also eventually lead you to be able to *think* in Greek, and that, to mix cultural mythologies a bit, is the Holy Grail. Several years ago when I was desperately trying to pass my French classes (huzzah for the pity D) I found that I had to stop myself from using French prepositions in daily conversation. This was the first sign of internalization- but since I suppressed it so hard, it never came to anything.
The previous poster was also correct that the more often you use the language- not just study it, but use it- the easier it will become. Good luck!