To make a payment in euros, you will need to convert your US dollars into euros. This can be done through a currency exchange service or through your bank. Once you have converted your funds, you can then write a cheque in euros to the recipient in Eire.
Alternatively, you may also be able to use a debit or credit card that is denominated in euros to make the payment. This can be a more convenient option, as you will not need to worry about converting your funds beforehand.