The three domains of practice offered by the Buddha are body, speech, and mind. Although not fully separate, they are worth examining distinctly.
There is a possibly apocryphal story from IMS (a retreat center in Barre MA) about a yogi on an extended retreat. Early one morning, when the breakfast cook arrived to start the oatmeal, she found a yogi with his hand in one of the food bins in the kitchen. Surprised to see the cook, the yogi said, "I was looking for a spoon."
We can laugh, but how often do we notice that our own speech is not perfectly aligned with our actions (or thoughts) in the moment? We may not be so skilled at speaking straightforwardly, finding instead that we hedge the truth, omit details that would make us look bad, or otherwise shade the language in our favor.
Looking at the suttas, we find this interesting reference from Iti 4.112 ("The World"): “As the Tathagata says, so he does; as the Tathagata does, so he says: therefore he is called the Tathagata." (Tathagata is a word for the Buddha). This quote says that perfect alignment between speech and action is a quality that characterizes Buddhas. Thus, we can realize that we may have a ways to go, and we may need some guidance on how to get our speech aligned.
For more detail on this topic, here is a talk I gave about it at Insight Meditation Center.
Two practices that provide excellent background and preparation for working with speech are mindfulness of the body and listening practice. It is amazing how much information is contained in the body, and how grounding it can be to rest the attention in the feet or belly while speaking. It is also worthwhile to attune to tensions in the body with the aim of relaxing them if possible. If you are not able to sense your body while speaking, begin by consciously practicing feeling your feet on the ground. As for listening practice, there are many resources out there for learning mindful or heartful listening; these are worth pursuing if it's of interest.
Moving on to speech practice, Right or Wise Speech is said to have four key qualities: It is true, beneficial, timely, and spoken with a heart of kindness. Note that "true" means both factually true and "true to the moment": It should express something appropriate to the situation, which often means it must be spontaneous. Although there are times when planned speech is necessary, often things that we have thought of ahead of time become conceptual and not quite "true" to the situation.
The Buddha was once asked (MN 58§9): “…when [people] go to the Blessed One and pose a question, has there already been in the Blessed One’s mind the thought, ‘If they come to me and ask me thus, I shall answer thus’? Or does that answer occur to the Tathagata on the spot?”
The Buddha replied, “It occurs to me on the spot.”
As with the domain of the body, there are two ways of practicing: Taking deliberate action to create wholesome or mindful speech, and observing our "natural" speech to feel how well it aligns with our intentions, the truth of the moment, etc. Both are useful modes of practice. In the second case, as with the body, it may take some practice to be able to speak and observe the speech carefully at the same time. It is well worth learning this skill. In the end, we want to speak spontaneously, like the Buddha, not relying on a mental filter that checks for the qualities of "right speech."
Sometimes people think that spiritually correct speech must always be pleasing to the listener -- in other words, it should be "nice." There are actually some suttas that say this. But other suttas place "pleasant speech" in the realm of timeliness. From MN 58§8: “In the case of words that the Tathagata knows to be factual, true, beneficial, but unendearing & disagreeable to others, he has a sense of the proper time for saying them.”
Teachers may say "unendearing" things in order to shock a student out of a complacent or deluded way of thinking. One teacher talks of a time when he was a beginning Dharma student, on his first retreat. He told his teacher in an interview that he thought he was the only person on the retreat who was not being mindful. She looked at him coolly and said, "What makes you so special?"
In the end, we are asked to speak with wisdom and compassion. That is all that "right speech" really points toward.