An economy of any scale running purely on one commodity is as impractical as one running purely on ad hoc barter.
Using a rare material like gold is most directly useful for large transactions, where it's practical for a detailed assessment of the value to be made independently by each party. With gold, this assessment is an objective one of weight and purity, with the only arguments being about how to prove the purity. With diamonds, some appropriate scale of "quality" would be needed, which might lead to more disputes along the lines of "I asked for 200 grade 5 diamonds, some of these are clearly grade 4".
For smaller every day transactions, people do not want to be constantly assessing the value of every token themselves, they want someone to vouch for it. That's where coins come in: the government, or some other body, assesses the weight and purity of a piece of metal, and stamps it as a particular value. Most transactions proceed based on the authenticity of the stamp, not the weight and purity of the metal.
Moreover, the minting authority standardises that value, so that you don't have to work out what "23g at 80% + 19g at 70%" gives you, you just count coins in a few fixed denominations. With precious metals, these denominations can be arbitrarily small, just by adding more base metal - a single coin containing 80% gold by weight can be recast into 80 coins of equal weight but only 1% gold.
So, someone will need to certify the diamond. Engraving it with a hallmark is tricky, so perhaps you embed it into a minted coin, as Russell Fox suggests. Rather than just measuring purity, as with precious metals, the mint will be acting as arbiter of disputes about quality, and how two small diamonds relate in value to one medium-sized diamond.
They'll also need a simplified system of denominations, so that the coins are freely interchangeable. A challenge there is that diamonds can't be arbitrarily divided and recombined like molten metal, and their value isn't linear - two 5-carat diamonds are not worth the same as one 10-carat diamond. Most coins will therefore probably have multiple diamonds embedded, rather than fractions as would be the case with gold coins. Whether that's practical will depend on how common diamonds are.
At this point, your diamond economy quickly starts looking like real economies in history - the controller of the mint can start inflating the currency by over-certifying diamonds; and smaller denominations of coin can be minted with no diamonds at all, valued because the mint promises to exchange them at a certain ratio. The "diamond standard" eventually becomes a largely abstract measure for comparing the validity of coins minted by different authorities, complicated by the more subjective nature of "diamond quality" as opposed to the purity of a precious metal.
In other words, you have bags of diamonds in Fort Knox to facilitate international trade, but you don't have someone handing a bag of glistening gems to their local baker in the morning, they hand over copper pennies just like in our world.