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Here are some tips given by Srila Prabhupada, Srila Bhaktisiddhanta Maharaj and Srila Bhaktivinode Thakur on how to chant the holy names of God without offense.

1) Nectar of Devotion – Chapter 13
Chaitanya Mahaprabhu has recommended that everyone chant the Hare Krishna mantra just to cleanse the dust from the heart. If the dust of the heart is cleansed away, then one can actually understand the importance of the holy name. For persons who are not inclined to clean the dust from their heart and want to keep things as they are, it is not possible to derive the transcendental result of chanting the Hare Krishna mantra. One should, therefore, be encouraged to develop his service attitude toward the Lord, because this will help him to chant without any offense. And so, under the guidance of a spiritual master, the disciple is trained simultaneously to render service and at the same time to chant the Hare Krishna mantra. As soon as one develops his spontaneous service attitude, he can immediately understand the transcendental nature of the holy names of the maha-mantra.

2) A Second Chance (book on reincarnation) - chapter on Atonement
Sridhara Svami says, “Chanting the holy name of Narayana, Lord Hari, not only counteracted all of Ajamila’s sinful reactions, but it made him eligible to become liberated, and therefore at the time of death he was transferred to the spiritual world.” Chanting the holy name of Krishna counteracts all sins - more than one can possibly commit. Because Ajamila had chanted the holy name without offense, and because he chanted sincerely, he completely freed himself from all sinful reactions. This is why we are stressing the chanting of Hare Krishna so much. Pious activities, austerity, sacrifice - everything is done simply by chanting the Hare Krishna mantra. There is no need of any other process of yoga, atonement, or austerity and penance. Simply by chanting without offense one achieves the results of all other ritualistic performances prescribed in the Vedas. The key to successful chanting, therefore, is to carefully avoid sin altogether. By chanting the Hare Krishna maha-mantra, a person is freed from all reactions to his previous sins, and if he avoids sinning again, he very quickly becomes an advanced devotee. His mind becomes fixed upon the lotus feet of Krishna. If we regularly chant Hare Krishna without offense, we shall remain free from all sinful reaction, and our attachment for the Supreme Lord in devotional service will increase.

3) Srila Bhaktisiddhanta Maharaj, on pg. 27 of the book called Siksastaka, says: There is indeed a way to escape the steely clutches of namaparadha. One must recognize and isolate the offences conscientiously avoiding them, he must then chant continuously.

4) Wear the “trnad api sunicena” mantra around one’s neck. Which means always be aware of these four things while chanting and serving in this world (be very humble,  more tolerant than a tree, devoid of false prestige, and offer proper respect to all).

5) Chant the Siksastaka daily because this will develop one's love for Krishna.
If anyone recites or hears these eight verses of instruction (Siksastaka) by Sri Chaitanya Mahaprabhu (daily), his ecstatic love and devotion for Krishnaincreases day by day. (CC. Antya 20.65)