There are a number of Pali terms that are roughly equivalent to the English idea of "letting go". In the Third Noble Truth you find the terms cāgo paṭinissaggo mutti anālayo, which are all variations on the theme of letting go. Cāga means something like "giving up," and it is closely associated with generosity. Paṭinissagga is usually rendered as "relinquishment" and often has a deeper sense of letting go, as in the more profound stages of the path. Mutti is related to vimutti, “freedom” or “liberation”. I suspect this simply refers to not holding on to things, but setting them free, that is, letting go. Anālayo means something like “non-settling” or “non-clinging”. (It is related to Himalaya, which means the place where the snow settles, hima being a term for snow.)
There is also the word vossagga, which is related to paṭinissagga. Vossagga is found in the compound vossagga-rato, “delighting in giving up,” which specifically refers to delighting in generosity. Vossagga is found in an interesting definition of the samādh'indriya (SN48:10): vossagg'ārammaṇaṃ karitvā labhati samādhiṃ labhati cittassa ekaggataṃ, “having laid a foundation through letting go, one gains samādhi, one gains unification of mind.”
The above ties in with nekkhamma, which is part of the second factor of the noble eightfold path. Nekkhamma is the opposite of kāma, and as such it refers to the giving up of, the letting go of, sensuality. This is an important part of the path to samādhi, and it is therefore related to the vossagga just described. The letting go of sensuality is really only possible by seeing its danger or the suffering it engenders. This is described well in the suttas by the triad of assāda, ādīnava, and nissaraṇa, gratification, danger, and escape. It is by understanding that the danger (or suffering) of something is greater than the gratification that one seeks and finds the escape, which again relates closely to letting go. The danger in sensual pleasures are described at length in MN 13 and MN 54.
An interesting use of paṭinissagga is found in the Ᾱnāpānasati Sutta (MN 118) where it is the final stage of insight, following on from anicca, virāga, and nirodha. The three latter terms can be regarded as describing different degrees of impermanence, and paṭinissagga is then the relinquishment that is a consequence of seeing impermanence deeply. This fits with the general progress of mental development whereby samādhi leads to yathā-bhūta-ñāṇa-dassana which in turn leads to nibbidā which leads to virāga which leads to vimutti (AN 10:2). Yathā-bhūta-ñāṇa-dassana, "knowledge and vision according to reality," is seeing the aggregates in terms of the three characteristics. Nibbidā is the disenchantment, or perhaps better the aversion or repulsion, that comes from seeing suffering and non-self deeply. You literally turn away from the world, from the five khandhas, which of course is a strong form of letting go. This then leads to the fading away (virāga) of craving and then liberation from it (vimutti). This is really a description of letting go in the advanced stages of the path.
