Read your text aloud. You don't have to write like you speak, but reading aloud changes your perspective. Awkwardness will jump out at you.

Reading aloud is one way to get some distance, to separate the piece from your memory of writing it. Putting it aside for a day or, better, a week does the same thing. I find that reading a printed copy helps me see things I don't see on a screen. Can you find other tricks? Richard P. Gabriel tells the story of one writer who would tape his work to a wall, go to the other side of the room, and read it through binoculars.

I hardly ever read my text aloud without remembering an incident from my days as an English major. In one class, we had to write a poem. Other people read them aloud. When someone read mine, I discovered that what sounded OK when I read it sounded awful when he did. There were places where I slowed down, sped up, or placed emphasis and he did not. He didn't because there were no cues in the text to tell him to do that. All the cues were in my auditory memory or imagination.

Recently I've been experimenting with having my Powerbook read the text to me (Program -> Services -> Speech). "Vicki's" rather odd intonation helps me find awkwardnesses that I don't otherwise notice. She's not a replacement for my own reading, but I think it's worth listening to her reaction.