It's a bonus not a bribe

Although some trainers feel that food should not be used, it is a very useful, practical tool for helping the dog to learn quickly what it is you want them to do.  

If you use food, to get the dog to do something they have already learned how to do, then you are using food as a bribe. e.g. showing the dog a treat before you ask the dog to sit. You are also using food as a bribe if you offer it as a payoff to stop the dog from doing something undesirable. e.g. giving a dog a chew bone to stop excessive barking.
 

A dog can earn a reward. A bribe is showing the dog the reward ahead of time.  A reward is a surprise payoff for doing good work.

However, you can use food in two other ways that assist effective learning and reinforce good behavior for a lifetime. You can use food to lure or show the dog what you are teaching them, and to reward as a paycheck for doing good work.

If you intermittently surprise the dog with a food reward for responding quickly to your requests, then you are using food as a bonus. Think of your praise as the salary and the food treat as the bonus. This helps to reinforce a dog that is attending to your desires, awaiting your intentions, deferring to your needs, and responding to what is learned. 

Other training rewards can also be used such as squeaky toys, balls, praise, petting, and any other thing the dog perceives as a reward - but food will probably provide the most motivation.

Think of your praise as a salary and food treats as bonuses.

Using food as a teaching tool

To use food as a teaching tool, use it as a lure. Hold the treat right at the dog's nose or lips and allow the dog to attempt to nibble it. Then, move the food to  position the dog correctly. For example, if walk backwards, moving the food forward, and the dog follows, say the word, "come" to teach the dog that walking to you is what you want. Lift the food slightly up and then back so that the dog's head tilts up and back until the dog naturally sits. As the dog sits, say the word, "sit."  Luring to show the dog exactly what you want and pairing that action with the desired word or cue is a terrific use of food. It is much faster than placing the dog physically into position because with luring, the dog is thinking and figuring out the action on his own. Using petting as a reward often confuses dogs as they try to read our body language and respond to our movements. Dogs will often learn hand signals before they learn what words means.
Some dogs work better for a special toy than for food. Luring can be done using the special toy by positioning it while withholding it until the action is completed, then releasing the toy as the reward.

The advantage of food is it can be used in three ways.  First as a lure, second as a reward, and third as a reminder.