Strictly speaking, whether prayer works is an empirical question. Pray for some, not others, see who gets better. Now, I realize there are possible theological implications to this kind of study, but various researchers have done it.
Recently David Hodges, of Arizona State, did a meta-analysis of 17 empirical studies "the effect of intercessory prayer among people with psychological or medical problems," and he found an overall positive effect.
His conclusion: "prayer is effective." Cool!
Thanks to Gary Sweeten for point out this article.