Let's start with a disclaimer, which is typically a bad idea. I do not have much artistic talent, at least in the area of graphics or visual media. I am often amazed by people who do, and a few of those people are my friends.
Check out the Netdiver Best of the Year 2005 awards for web site design. It's a great example of why you don't want designers making too many decisions about web site design. The sites on the list are graphically rich, and there are some innovative elements, but more importantly, they are hard to use, hard to understand, slow to load, and dogged by other difficulties.
In general, the sites scream out, "Look what I made the web do!", rather than trying to get across a more useful message. Maybe I am too much of a left-brain thinker, but those sites don't work for me at all.
If I was looking for a designer, I'd want the sites I visit to describe the services they provide. Most of the sites on the list do don't do that well, if at all.
I'd want the site to include a portfolio. Most of the sites do that, but the portfolios only show the work. They do not describe what the client was hoping to achieve, whether those goals were met, etc.
Take a look, and decide for yourself.