We've had the toast done several ways. Honestly, it is completely up to the happy couple. The "traditional" toasting is done by the Maid-of-Honor & Best Man and, if you would like, the fathers. I recently had a wedding were the bride and groom decided not to have any toasting at all.
Limit the time that people are going to be making a speech will help, ask those important to you to prepare a statement that they can share, because a rambling story from a drunk friend or family member just causes tension.
I agree with the other comments; you don't have to open the mic to your guests for general comments. I would like to add a point & a tip. Point: Traditionally, the Father of the Bride gets to walk his daughter down the aisle, give her away & give a toast. This often, understandably, leaves the Mother of the Bride feeling overlooked. Tip: The toast is a great time for your parents to share in blessing you both. As long as you have a loving, or even just cordial, relationship, invite them up together (or individually, in the case that they're no longer a couple), and allow them each time to express their joy regarding this new step in your lives. Same goes for the groom's parents.
P.S. It's ok to suggest a time limit.
At my wedding we had the Best Man, Maid of Honor, sister of the groom, and Father of the Groom do toasts and that was it. No one else seemed to care and we were fine with keeping it short. (sister of the groom only did a speech to get back at my husband for the speech he gave at hers! hehe)
It's your wedding, talk with your friends and family ahead of time and let them know that only X people will be giving toasts. ;)
You can decide ahead of time who will do toasts at your reception. The Best Man is the most typical toast that people expect to hear at a wedding. Other traditional toasts can be given by the Maid of Honor and parents of the bride or groom.
If you are concerned that unwanted or too many toasts will occur, you can give your master of ceremony a pre-selected list to announce who will be toasting. Frequently this can be done by your DJ or entertainer. Then they can return to playing music immediately after the last toast. Planning this ahead of time will take the pressure off of you and you won't need to handle this during your reception. You can let your bridesmaids and groomsmen know about your wishes beforehand too.