This is a widely-accepted myth. You only hear about the failures because they make good copy. A few examples of successes:
(rdbpc = randomized, double blind, placebo controlled)
Vitamin D has been shown to reduce cancer 35% in a rdbpc trial (taken with calcium the reduction was 75%).
Vitamin D has been shown to prevent the flu (25% type A only) in 2 rdbpc trials.
Vitamin C has been shown to reduce the duration and severity of the common cold (but not its frequency).
Selenium has been shown to reduce cancer risk (incidence or death 27-58%) in a few rdbpc trials.
Selenium (with CoQ10) lowered the risk of dying from heart disease 45-50% in a rdbpc trial.
I could go on for pages, but instead refer you to the
University of Michigan Health Library. The references are rated based on quality: size of group, type (observation < epidemiology < clinical trial < rdbpc trial), etc. A lot of good studies aren't referenced here, but it's a good start.