Not the most exciting of topics but it does seem to come up a lot. It can be very simple and can also be made very complicated. I always prefer simple.
When I began printing I carried everything to the kitchen sink and attacked my kit with washing up liquid, water and a lot of elbow grease. This resulted in a couple of nice blue stains on the stairs carpet as a blue inked roller bounced down them! I'd then leave everything on the draining board to dry. I have a tiny kitchen and there's not really room to leave anything else other than kitchen equipment out.
I am a member of Linocut Friends group on facebook and the 'clean up' question comes up often. I decided to experiment with some of the suggestions. This is what I have come up with. It works well for me as a home printer.
I use Caligo Safewash Inks, other inks may need a different method.
Things you'll need
Scrap paper or newsprint
Roll your rollers on scrap paper to remove as much ink as possible. I used to do this on junk mail but now I do it on plain newsprint and save it for wrapping paper. Make sure you roll the edges of the roller too as ink often builds up here.
Scrape as much ink off your glass plate (or whatever you roll on) as you can. If there's enough to use again fold it into some greaseproof paper or a scrap of plastic. Tape it up and pop it in a jam jar.
Wipe your scraping tool on scrap paper.
Blot your lino on more newsprint - for more wrapping paper!
Now squirt some baby oil onto a rag and wiped all surfaces clean. Keep changing the rags if they get saturated with ink/oil.
Pull the rag through any gaps in your roller. I do not use expensive rollers, but if you do these may unscrew and have specific cleaning methods.
Take a clean rag and add a little Zest It. Wipe over everything.
Orange/red rollers cleaned on newsprint and then over printed with a black block.
I hand wash the rags and re-use, I have a friend that uses them as fire lighters but that may well be rather dodgy!
Block blotted on brown paper used as wrapping paper