Pure rectangular grid design is unsurprisingly a very popular website design strategy. It allows clean simple easy to follow patterns. However, it takes good design skill to stand out from everyone else who is doing this.
If appropriate you can stand out with a more "organic" style. Although you will probably end up building this using some from of grids the result can give a more "natural" feel. It is important in a less formal design style to use colour and contrast to guide the user appropriately and to not make the style distracting and confusing. Another technique to allow this naturalistic style is animation.
Appropriate animation is probably one of the most under used techniques, perhaps because it is difficult to pull off, but when it is done well, such as in the transitions in iOS and Windows 7 phone then it can really help guide context and application understanding.
Found a nice little demo that can help with a useful transition highlight effect, which can be seen to help guide context... Hopefully animation (not the horrid flash / gif stuff of yester year) can improve with helpful frameworks developed in HTML5, well certainly nice guides and demos will help with this.