Although the practices in current ideas of mindfulness, such as breathing exercises and the particularly Eastern metaphysical ideas about the self (like in, for example, Reiki) wouldn't specifically have been invoked by anyone in ancient Greece, there are strong parallels with the theorising of the Stoics (SEP Article), and in their conception of Philosophy itself. Consider, for instance, the following cool quote from Cicero:
But there is one method of healing both distress and all other diseases of the soul, namely to show that all are matters of belief and consent of the will and are submitted to simply because such submission is thought to be right. This deception, as being the root of all evil, philosophy promises to drag out utterly.
The Stoics argued that the correct approach to Passions, being emotional states of being where the mind is specifically moved to act in a way contrary to reason, nature or virtue, is to recognise them through a life of practiced philosophical self-awareness and to thereby be equipped to resist their influence. This seems very much in line with eastern ideas of mindfulness.