Note: I don't dislike the term "alpha" in and of itself. It essentially means a higher-ranking animal, one who leads the group. I dislike the connotation it now implies (thanks to a few choice trainers) - forceful, dominating, alpha-rolling, etc.
I definitely agree that we need to be the leaders of our dogs. However, being a leader, or even being "alpha", if you will, is NOT being physically dominating. It has been shown that in wolf behavior (where this alpha-dominance theory originated), the higher ranking alpha/leader wolves do not physically "roll" the other wolves - there is no force. The other wolves know their place, and they offer the roll as an appeasement gesture.
Being a good leader is controlling the resources. And controlling the resources is not things like not allowing the dog on the furniture or the bed, making the dog eat after people, making the dog walk through doors after people, etc. Controlling the resources means making the dog understand that all good things in life come from people. Food, water, toys, attention, everything. If you want food, you have to sit first. If you want to go for a walk, you have to walk nicely and not pull. If you want to play fetch, you have to drop the ball so I can throw it again.
Being a good leader is also making it so the dog listens because it wants to please you, not making it so the dog listens because it's afraid of what will happen if it doesn't. Reward positive behavior with treats, praise, attention, toys, whatever works for you and your dog. Ignore the negative behavior, or, better yet, ask for/teach a behavior that you WANT instead. For example, if the dog is chewing on the table leg, don't yell/scream/hit the dog. Hand it a chew toy or a bone, and praise it for doing the right thing.
Every person is capable of training a dog and being the leader. Every dog is capable of being trained and being led. However, not every person is able to be physically dominating, and not every dog is able to be physically dominated.