A lot has been written about making written requests, such as for a meeting. Based on my inbox, a lot of good advice is being ignored. So, here's my two-cents on the subject:

Good work often starts with research. I recommend that you save the meeting requests and general sales letters you like in a dedicated folder and, before writing your next request letter, review them for tips and inspiration. Pay particular attention to what you like about the wording, layout, flow and tone of your favorite letters.

Staying with research, investigate the companies and people your soliciting. There's an amazing amount of information available today. Google and LinkedIn are two of my primary sources of client research. In 15 minutes you can typically learn a person's professional history, job title, interests, status in their industry  and many even identify some mutual friends. This will help with the style, tone and personalization of your letter.

Create an outline for your letter as follows:

1. Grab your reader's attention. Begin with an interesting fact, important question, comment on a current event or something personal, such as congratulations for being named Person of the Year.
2. Then transition your reader into the purpose of your letter - introduce your company, request an appointment, or a free offer, for example. Be sure to connect your request with your grabber from (what will be) paragraph one. Your transition is key, it must be both smooth and brief, people are busy. See next bullet.
3. In today's hurry-up, get-to-the-point world, many people switch the first two bullets of their outline. they begin letters by coming right out and stating, "The purpose of this letter is to request a 30-minute appointment to..." and then write their grabber. I prefer a subtler approach but will use the direct approach if I know the reader well.
4. Insert numbers or testimonials that back up your claims next. A list of indented bullets works well for either. People like reading lists of tightly written facts - summaries.
5. Your fourth section/paragraph is your call-to-action. Tie your attention grabber or your reader's self-interest into the benefits of responding to your solicitation. I don't sell actual products so I often use mutual gain as my call-to-action: "I propose a 30-minute meeting to better understand your business objectives, review our capabilities and determine if we can help one another achieve our  goals..."
6. Your final outline point let's your reader know that your letter is one of a series of contacts, that there are more to come. Explain that you will be following up by voicemail, email or both. Basically, you're telling them "you're not going away, so let's have our conversation and see where it leads."

Now it's time to fill in the blanks and edit. Go back to each section of your outline and write 2-4 complete sentences. Often this is as easy as writing a topic sentence, inserting your outline copy and, then, writing a closing sentence that introduces your next point (paragraph) and compels the reader to keep reading.

With my writing completed I move onto the editing process. I complete three rounds of edits. My first edit is for grammar, my second is for flow and my third is for appeal. The final edit, for appeal, answers the question, "Would I respond to this letter?" If I would, then I'm done. If, however, there are awkward transitions, facts that don't seem to fit, or anything else that makes my letter weak, then I keep writing and editing until they are corrected.

The person receiving your letter is busy. They are looking for reasons to throw your letter away. It's your job to give them reasons to keep it and respond.

Mail your letter and begin following up within a week.

Final thought:
Some may argue that Bullet #6 above is aggressive. I'll concede that point. However, if you've done your homework and are only contacting people you honestly believe you can help, then why be shy? You're trying to earn a living and help others along the way; what's wrong with that? I'm not looking to waste my time or anyone else's on silly meetings nor am I looking to sell anyone something they don't need or want. I have pride in what I do and so should you.

In summary, this comes down to professionalism and character, use a strong doze of both in everything you do and trust that positive results will follow.

Steve