Actually there is one that occurs to me, somewhat the reverse of one that English speakers can use to get the French u.
Make the u sound in "puro". While doing this, move the lips to make the i sound in "mi" ("unround" the lips, if you would), without (and this is important) moving the tongue. This should come pretty close. You will find that the main issue is to avoid pulling the back of the tongue up to the roof of the mouth as you do with the "mi" sound; the degree of pressure of the sides of the tongue against the sides of the mouth can vary more.
Another way to go at it is to make the "mi i" sound and attempt to relax and drop the back of the tongue from the roof of the mouth.