Depending upon religious, cultural, or economic backgrounds, couples who are planning to get married tend to feel (almost subconsciously, perhaps) that their musical preferences are fairly irrelevant to what sort of music needs to be played at their ceremony. They don't consider the fact that there is no hard and fast rule that states that all musical pieces played must be either The Wedding March or Canon in D, and it all must be played on a single piano or organ.

Those songs, played in that manner, are very traditional and very simple. They're certainly functional as musical backdrops for your ceremony, but they aren't required or even necessarily desired by everyone.

Here are some options you may want to consider:

Choral-only performances - Vocal performances, if you have access to a choir, can be either a hauntingly beautiful experience (think of the choral performance from My Best Friend's Wedding, for example), or a get-out-of-your-seat experience that will uplift your soul - and everything between. Vocal performances have the capacity to sound more compelling, intimate, and, wellpersonal. Definitely consider finding a local choir to at least get a sample of what they can do; most choirs are willing to perform for free, or at most, a nominal fee.
Strings - A violinist here, a cellist there, and you've got yourself something special. Nothing says "romance" like strings, and you will find that your ceremony has taken on a very romantic feel in the classical sense. For the ultimate in traditional musical beauty, the employment of a string section is a great way to go.
Pre-recorded music - Obviously, you have access to an infinite variety of musical styles right at your fingertips if you're willing to compile a mix CD; some people think that wedding music must always be "live," but that really isn't so; The sheer variety of choices available if you go the pre-recorded route make this a strong sell for couples who either don't wish to confine their musical selections to a single type, or who are working off of a budget that doesn't allow for live performances.
Piano / Organ - If you do feel safe sticking with a pianist / organist, understand that you can request other songs besides the aforementioned two in the above paragraph. Most musicians would be happy to play anything you would request, as long as they have sufficient time to feel that they've mastered it. Don't expect most musicians to be willing (or able) to instantly whip up an "improv" performance.

No matter what you end up choosing, there's no need to stick with musical selections that you find dull or over-done just for the sake of tradition. There is more music you could potentially use in your wedding than you could listen to in several lifetimes - don't be afraid to experiment with something you like just because it hasn't been conventionally used before!