Each State Different
Because each state's requirements are different, you should investigate your formation state's requirements for dissolution before making a decision. You can start the research process by going to State Research (from the SmallBiZ.com Home page) to find your state's website location for more details.
Who to Notify
After you have formally dissolved your LLC, you must notify those that may have an interest in your LLC's activities. Besides the obvious of settling all of the LLC's debts/liabilities (usually required before dissolution of the LLC) and closed out your bank account, here are some other people/entities you'll need to notify:
If you have obtained an EIN and you have previously filed a tax return listing your LLC's activities, or you intend to list the LLC's activities (receiving income and or expenses) on the next tax return, you can indicate the dissolution in your next tax return. Otherwise, if your LLC has had no activities so far, you need to send a letter, on the company's stationary, and listing the LLC's EIN, stating that the company is dissolved and no longer conducting business. Send this letter to the address on your EIN acceptance letter from the IRS.
Other State Agencies
Although the agency that forms and dissolves LLC's would be obviously privy to your LLC's dissolution, do NOT assume all other state agencies will get the memo. In fact, you should assume otherwise. You will also want to notify your state's Dept of Revenue (whomever the tax body is), and any other state departments, boards of licensing, or other agencies, that might have been originally notified as part of your registration.
Your LLC's registered agent, if it is not you, needs to be told of the LLC's dissolution. Do this in writing, or as requested by your agent.
If you have other vendors or companies/people that the LLC has agreements with, or credit, you might want to make sure that they are notified as well.