People (myself and fellow commentor included) recommend reading up on the konmari method because it is a simple process that helps you deal with your relationships with things and how to increase mindfulness in ownership.

Since you wish to dedicate yourself to zen practice by getting rid of as much as possible, it sounds like diving into the deep end of the pool. Before you do so, even if you have very little attachment to your things, keep in mind that minimalism is a tool for your spiritual development and not the goal: use it to your advantage when creating a space that is minimal and learn from it as you go through the process, but do not dedicate yourself to it at the expense of anything you find special to you or cause to to obsess over maintaining the least amount of things over your own peace of mind.

If you are getting rid of something and that particular thing feels like a 'sacrifice' to remove, do not simply throw it away in your pursuit of zen. Stop and consider why it feels like a sacrifice and why you are throwing it away. Simply powering through the decluttering process is ignoring the now, when paying attention to the now is an essential facet of Zen.

So cold turkey or weaning off is more of a matter of what time window you have available to the decluttering process. No matter which method you decide to adhere to, treat it as a ritual and a time for reflection - focus on the now.