The play is set in Messina, Italy, in and around Leonato's house (who the text makes clear is the Governor of the town). The soldiers, led by Don Pedro of Aragon, seem to originate in Aragon (in Spain) but their reason for being in Messina (we know they have just won a war of some description) is unclear - is it a personal visit, or have they been billeted?
Shakespeare's sources for 'Much Ado' seem to be drawn largely from classical or knightly romances (Ariosto and Bandello in particular are often cited as sources for the play) and so perhaps Shakespeare saw the play in similar terms. But unfortunately, he left no detailed information about the period in which he wants the play to be performed - and the absence of historically tracable characters (i.e. in 'Henry V', we can just rehearse the history of the real Henry V) makes it a difficult decision to make.
Scholars often believe that Shakespeare's plays were always performed in the modern dress of the day (a supposition fuelled by teh Peacham engraving of Titus Andronicus, which shows actors in versions of Elizabethan costume) - and so one could argue either that the play should be performed in Elizabethan dress, or in the modern dress of OUR day. Both could be seen as being true to Shakespeare.