According to legend, Shani is a devotee of Lord Shiva. As mentioned in the "Navagraha Pidahara Sthothram" of "Brahmanda Purana", anybody who chants the following sthothra is relieved from all the ill effects of Shani.
Suryaputhro Deerghadeho Vishaalaakshah Shivapriyah |Mandachaarah Prasannathmaa peedam harathu me Shanih
||सूर्य पुत्रो दीर्घ देहो विशालाक्षः शिवप्रियः।मन्दचारः प्रसन्नात्मा पीडां हरतु मे शनिः ॥
The Shaneeswara Mantra In Tamil:
Sangadangal Theerpai Shanibhagavane
Mangalam Ponga Manam Vaitharul
VaiSacharavindri Saaga neriyilIcchagam
Vaazha Innarul Thaa Thaa
According to the Vedic astrology, in order to get protection against the bad effects of Saturn's transit, the following are the possible remedies during the important transits of Saturn - Kanta Sani (When Saturn transits through the eighth house from the natal moon sign), Sade-sati (when Saturn transits through the twelfth, first and second houses from the natal moon sign) -
Worship Mother Goddess Kali during Dark Moon
Worship Vishnu in the form of Lord Krishna and keep chanting 'Om Namo Narayanaya', 'Hare Krishna Hare Krishna Krishna Krishna Hare Hare..'
Pray to Sree Hanumanji in His Supreme Cosmic Form, which is the Form in which He shook Shani Bhagavan off of His back and the form in which He received divine blessings and instructions from Surya, the Sun God.
A common mantra for drawing the support of Shani Bhagavan is: Om Sham Shanaischaryaye Namah.
The following is another mantra for propitiating Shani:
Neelaanjan samaabhaasam raviputram yamaagrajam,
Chaayaa-maartandam-sambhootam, Tam namaami Shanaishcharam.
Some devotees chant 'Om Sham Shanaischaryaye Namah' 108 times daily. Also, if reading vedic textures are practised then it is suggested to recite Saturn's shloka from Navagraha Sukta: 'Om Shanno Devirabhishtaya Aapo Bhavantu Peetaye Shanyorbhisravantumah Shanaishcharaaya Namah.'
Perform thilabhishekam on sanitrayodasi, sanijayanthi (pushya masam bahula astami) and saniamavasya days.
Donate black till to a Brahmin (1 kg 900 grams)
Feeding till and jaggery mix to black cow (kapila govu) .
Fasting on Saturdays (only liquids) 6 a.m to 6 p.m (sravana masam must)