Text shadow is not a good choice for long sections of text as the differences between the contrast and brightness of the user's display can cause the effect to render the text hard to read or illegible. On headings, having a larger font size, this tends not to be as much of a problem.
Sites that do use shadow on non-heading text tend to do so with subtlety, with just a pixel distance. It can work well when a light shadow colour is used to help dark text stand out against a medium to light background or gradient, but I'd still be cautious about using it for large runs of text.
Text set on anything other than a plain background also risks illegibility. If a background image is used it should be subtle in colour and shape. Geometric patterns such as diagonal lines cause one's focus to run away from the text.
When targeting a general audience, which I'd guess your site is, it's best to take a restrained approach to ornamentation. Tastes vary, so seek to avoid offending the viewers sensibilities. Express the character of the site in the colour scheme, typography and logo. Avoid "in your face" backgrounds. Consider popular sites on the 'net: Google, eBay, Amazon, Facebook etc. Do any of these use bold patterned backgrounds or shadowed text? They are successful because they provide a useful service, which is far more important than prettiness.