According to Gutch, 1901: 61 in Yorkshire quaking grass (Briza media) was known as trimmling jockies or doddering dickies, and it was said:
A trimmling jock i’ t’ house
And you weeant hev a mouse.
‘Dried in bunches, with its brown seeds on a tall stem, it was commonly stuck on the mantel-shelf, as believed to obnoxious to mice.’
This belief is also mentioned in a 1938 novel by Elizabeth Goudge, set in Oxford during the reign of Elizabeth I (1558-1603):
‘He learnt … that quaking grass, gathered and brought into the house, keeps away mice; an interesting fact and of value to those to whom a cat in the house is anathema.
Put a Tumbling Jockey in
June in your house
And he’ll rid you forever of
Every mouse.
Grace repeated the rhyme to him and showed him … quaking grass she had put in every room … and he promised that next year he would go out into the June fields and pick them for her.’
Was Goudge relating a local Oxford belief, a sixteenth-century belief, or had she adapted the rhyme given in Gutch, and also in Britten and Holland, 1886: 474, where it said to have been recorded in Whitby?
According to a report in the Royal Cornish Gazette in September 1805, hound’s tongue (Cynoglossum officinale), ‘gathered full of sap and bruised with a hammer’, will make mice and rats immediately leave barns and granaries [Pickering, 1995: 20]. Grigson [1987: 280] lists rats-and-mice as a Wiltshire name for hound’s tongue, possibly indicating that this belief was known elsewhere.
Finally, in a letter published in BBC Wildlife, 24(4): 94 (April 2006), a writer from Italy reported that a humane way to get rid of mice ‘is to plant rue Ruta graveolens in your garden and put some small branches of this in rooms overrun by mice. The smell will soon drive them away.’

References
Britten, J. & Holland, R., 1886. A Dictionary of English Plant-names.
Goudge, E., 1938. Towers in the Mist.
Grigson, G., 1987. The Englishman’s Flora.
Gutch, E., 1901. County Folk-lore, vol. 2.Printed extracts concerning the North Riding of Yorkshire, York and the Ainsty.
Pickering, I., 1995. Some Goings On!: A Selection of Newspaper Articles about Fowey, Polruan and Lanteglos District 1800-1899.

Thanks to Jill Lucas for drawing attention to Goudge’s novel.

Addendum. Walter Wade, in his Plantae rariores in Hibernia inventae (Dublin, 1804) notes:  ‘The Silesians are said to scatter the green leaves [of  dwarf elder, Sambucus ebulus] where their pigs lie in order to prevent some the diseases to which they are liable; certain it is, that the green leaves drive away mice from granaries.’

Image: house mouse, Mus musculus; Wiki Commons.

Addendum added 7 September 2024.